Pál Rakonczai, László Varga, András Zempléni Copula fitting to time- dependent data, with applications to wind speed maxima Eötvös Loránd University Faculty of Science Institute of Mathematics Department of Probability Theory and Statistics
P á l Rakonczai, L á szl ó Varga, Andr á s Zempl é ni Copula fitting to time-dependent data, with applications to wind speed maxima Eötvös Loránd University.
Dec 23, 2015
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Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
Copula fitting to time-dependent data with applications to wind speed maxima
Eoumltvoumls Loraacutend University Faculty of ScienceInstitute of MathematicsDepartment of Probability Theory and Statistics
Outline
1 Copulae
2 Goodness-of-fit tests
3 Bootstrap methods
4 Serial dependence
5 Applications to wind speed maxima
2
1 Copulae
bull C is a copula if it is a d-dimensional random vector
with marginals ~ Unif [01]
bull Existence (Sklarrsquos Theorem) to any d-dimensional
random variable X with cdf H and marginals Fi
(i=1d) there exists a copula C
H( x1 hellip xd ) = C ( F1(x1) hellip Fd(xd ) )
bull Uniqueness if Fi are continuous (i=1d)
bull Separation of the marginal model and the dependence
3
Elliptical Copulae ndash copulae of elliptical distributions
ndash Gaussian X ~ Nn(0Σ)
where Φ cdf of N(01)
ndash Studentrsquos t X ~
where tv cdf of Studentrsquos t distribution with v degrees of freedom
1 Copulae ndash Examples
4
d
u u
n
Ga dxdxeCtd
2
1)( 1
2
1)( )(
212
111 1
xxu
d
nv
tut ut
n
tv dxdx
vv
nv
Cv dv
1
1
2
2)( 1
21
)( )(
212
11 1
xxu
0Student vn
Archimedean Copulae Copula generator function
ϕ is continuous strictly decreasing and ϕ(1)=0
d-variate Archimedean copula
ndash Gumbel where
ndash Clayton where
1 Copulae - Examples
5
010)(u
d
iiu
1
1 )()( uC
1
1
log
)(
d
iiu
Gumbel euC
)ln()( uu 1
1
1Clayton 1)(
du
d
iiuC
1)( uu 0gt
6
1 Copulae - Examples
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gauss copula
075n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gumbel copula
25n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from t-copula
08n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Clayton copula
25n 1500
2 Goodness-of-fit tests in one dimension
1 Estimation of the model parameter
2 Goodness-of fit testa) Crameacuter-von Mises tests
bull Fn empirical cdf
bull F cdfbull Φ weight function
Anderson-Darling
b) Critical value ndash simulation1) Simulate a sample from the copula model Cθ under H0
2) Re-estimate by ML-method
3) Calculate the test statistics
Repetition and estimation of p values
Θ θθ CCH 00 C
)()())()(( 2 xdFxxFxFnT n
7
xFxFx
1
1)(
2 Goodness-of-fit tests in more dimensions
bull Probability integral transformation (PIT) ndash mapping into the d-dimensional unit cube
~H ~C for i=1n
bull Kendallrsquos transform (K function)
Advantage one-dimensional
ndash Example Archimedean copulas
where
))(()))(()((()( 111 tUUCPtXFXFCPtΚ ddd
8
nsObservatio
1 )( idii XXX nsobservatioPseudo
1 )(
idiiPIT UUU
tx
i
i
i xdx
dtf
1
tfti
ttK ii
d
i
i
1
1
1
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Outline
1 Copulae
2 Goodness-of-fit tests
3 Bootstrap methods
4 Serial dependence
5 Applications to wind speed maxima
2
1 Copulae
bull C is a copula if it is a d-dimensional random vector
with marginals ~ Unif [01]
bull Existence (Sklarrsquos Theorem) to any d-dimensional
random variable X with cdf H and marginals Fi
(i=1d) there exists a copula C
H( x1 hellip xd ) = C ( F1(x1) hellip Fd(xd ) )
bull Uniqueness if Fi are continuous (i=1d)
bull Separation of the marginal model and the dependence
3
Elliptical Copulae ndash copulae of elliptical distributions
ndash Gaussian X ~ Nn(0Σ)
where Φ cdf of N(01)
ndash Studentrsquos t X ~
where tv cdf of Studentrsquos t distribution with v degrees of freedom
1 Copulae ndash Examples
4
d
u u
n
Ga dxdxeCtd
2
1)( 1
2
1)( )(
212
111 1
xxu
d
nv
tut ut
n
tv dxdx
vv
nv
Cv dv
1
1
2
2)( 1
21
)( )(
212
11 1
xxu
0Student vn
Archimedean Copulae Copula generator function
ϕ is continuous strictly decreasing and ϕ(1)=0
d-variate Archimedean copula
ndash Gumbel where
ndash Clayton where
1 Copulae - Examples
5
010)(u
d
iiu
1
1 )()( uC
1
1
log
)(
d
iiu
Gumbel euC
)ln()( uu 1
1
1Clayton 1)(
du
d
iiuC
1)( uu 0gt
6
1 Copulae - Examples
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gauss copula
075n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gumbel copula
25n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from t-copula
08n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Clayton copula
25n 1500
2 Goodness-of-fit tests in one dimension
1 Estimation of the model parameter
2 Goodness-of fit testa) Crameacuter-von Mises tests
bull Fn empirical cdf
bull F cdfbull Φ weight function
Anderson-Darling
b) Critical value ndash simulation1) Simulate a sample from the copula model Cθ under H0
2) Re-estimate by ML-method
3) Calculate the test statistics
Repetition and estimation of p values
Θ θθ CCH 00 C
)()())()(( 2 xdFxxFxFnT n
7
xFxFx
1
1)(
2 Goodness-of-fit tests in more dimensions
bull Probability integral transformation (PIT) ndash mapping into the d-dimensional unit cube
~H ~C for i=1n
bull Kendallrsquos transform (K function)
Advantage one-dimensional
ndash Example Archimedean copulas
where
))(()))(()((()( 111 tUUCPtXFXFCPtΚ ddd
8
nsObservatio
1 )( idii XXX nsobservatioPseudo
1 )(
idiiPIT UUU
tx
i
i
i xdx
dtf
1
tfti
ttK ii
d
i
i
1
1
1
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
1 Copulae
bull C is a copula if it is a d-dimensional random vector
with marginals ~ Unif [01]
bull Existence (Sklarrsquos Theorem) to any d-dimensional
random variable X with cdf H and marginals Fi
(i=1d) there exists a copula C
H( x1 hellip xd ) = C ( F1(x1) hellip Fd(xd ) )
bull Uniqueness if Fi are continuous (i=1d)
bull Separation of the marginal model and the dependence
3
Elliptical Copulae ndash copulae of elliptical distributions
ndash Gaussian X ~ Nn(0Σ)
where Φ cdf of N(01)
ndash Studentrsquos t X ~
where tv cdf of Studentrsquos t distribution with v degrees of freedom
1 Copulae ndash Examples
4
d
u u
n
Ga dxdxeCtd
2
1)( 1
2
1)( )(
212
111 1
xxu
d
nv
tut ut
n
tv dxdx
vv
nv
Cv dv
1
1
2
2)( 1
21
)( )(
212
11 1
xxu
0Student vn
Archimedean Copulae Copula generator function
ϕ is continuous strictly decreasing and ϕ(1)=0
d-variate Archimedean copula
ndash Gumbel where
ndash Clayton where
1 Copulae - Examples
5
010)(u
d
iiu
1
1 )()( uC
1
1
log
)(
d
iiu
Gumbel euC
)ln()( uu 1
1
1Clayton 1)(
du
d
iiuC
1)( uu 0gt
6
1 Copulae - Examples
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gauss copula
075n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gumbel copula
25n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from t-copula
08n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Clayton copula
25n 1500
2 Goodness-of-fit tests in one dimension
1 Estimation of the model parameter
2 Goodness-of fit testa) Crameacuter-von Mises tests
bull Fn empirical cdf
bull F cdfbull Φ weight function
Anderson-Darling
b) Critical value ndash simulation1) Simulate a sample from the copula model Cθ under H0
2) Re-estimate by ML-method
3) Calculate the test statistics
Repetition and estimation of p values
Θ θθ CCH 00 C
)()())()(( 2 xdFxxFxFnT n
7
xFxFx
1
1)(
2 Goodness-of-fit tests in more dimensions
bull Probability integral transformation (PIT) ndash mapping into the d-dimensional unit cube
~H ~C for i=1n
bull Kendallrsquos transform (K function)
Advantage one-dimensional
ndash Example Archimedean copulas
where
))(()))(()((()( 111 tUUCPtXFXFCPtΚ ddd
8
nsObservatio
1 )( idii XXX nsobservatioPseudo
1 )(
idiiPIT UUU
tx
i
i
i xdx
dtf
1
tfti
ttK ii
d
i
i
1
1
1
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Elliptical Copulae ndash copulae of elliptical distributions
ndash Gaussian X ~ Nn(0Σ)
where Φ cdf of N(01)
ndash Studentrsquos t X ~
where tv cdf of Studentrsquos t distribution with v degrees of freedom
1 Copulae ndash Examples
4
d
u u
n
Ga dxdxeCtd
2
1)( 1
2
1)( )(
212
111 1
xxu
d
nv
tut ut
n
tv dxdx
vv
nv
Cv dv
1
1
2
2)( 1
21
)( )(
212
11 1
xxu
0Student vn
Archimedean Copulae Copula generator function
ϕ is continuous strictly decreasing and ϕ(1)=0
d-variate Archimedean copula
ndash Gumbel where
ndash Clayton where
1 Copulae - Examples
5
010)(u
d
iiu
1
1 )()( uC
1
1
log
)(
d
iiu
Gumbel euC
)ln()( uu 1
1
1Clayton 1)(
du
d
iiuC
1)( uu 0gt
6
1 Copulae - Examples
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gauss copula
075n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gumbel copula
25n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from t-copula
08n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Clayton copula
25n 1500
2 Goodness-of-fit tests in one dimension
1 Estimation of the model parameter
2 Goodness-of fit testa) Crameacuter-von Mises tests
bull Fn empirical cdf
bull F cdfbull Φ weight function
Anderson-Darling
b) Critical value ndash simulation1) Simulate a sample from the copula model Cθ under H0
2) Re-estimate by ML-method
3) Calculate the test statistics
Repetition and estimation of p values
Θ θθ CCH 00 C
)()())()(( 2 xdFxxFxFnT n
7
xFxFx
1
1)(
2 Goodness-of-fit tests in more dimensions
bull Probability integral transformation (PIT) ndash mapping into the d-dimensional unit cube
~H ~C for i=1n
bull Kendallrsquos transform (K function)
Advantage one-dimensional
ndash Example Archimedean copulas
where
))(()))(()((()( 111 tUUCPtXFXFCPtΚ ddd
8
nsObservatio
1 )( idii XXX nsobservatioPseudo
1 )(
idiiPIT UUU
tx
i
i
i xdx
dtf
1
tfti
ttK ii
d
i
i
1
1
1
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Archimedean Copulae Copula generator function
ϕ is continuous strictly decreasing and ϕ(1)=0
d-variate Archimedean copula
ndash Gumbel where
ndash Clayton where
1 Copulae - Examples
5
010)(u
d
iiu
1
1 )()( uC
1
1
log
)(
d
iiu
Gumbel euC
)ln()( uu 1
1
1Clayton 1)(
du
d
iiuC
1)( uu 0gt
6
1 Copulae - Examples
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gauss copula
075n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gumbel copula
25n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from t-copula
08n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Clayton copula
25n 1500
2 Goodness-of-fit tests in one dimension
1 Estimation of the model parameter
2 Goodness-of fit testa) Crameacuter-von Mises tests
bull Fn empirical cdf
bull F cdfbull Φ weight function
Anderson-Darling
b) Critical value ndash simulation1) Simulate a sample from the copula model Cθ under H0
2) Re-estimate by ML-method
3) Calculate the test statistics
Repetition and estimation of p values
Θ θθ CCH 00 C
)()())()(( 2 xdFxxFxFnT n
7
xFxFx
1
1)(
2 Goodness-of-fit tests in more dimensions
bull Probability integral transformation (PIT) ndash mapping into the d-dimensional unit cube
~H ~C for i=1n
bull Kendallrsquos transform (K function)
Advantage one-dimensional
ndash Example Archimedean copulas
where
))(()))(()((()( 111 tUUCPtXFXFCPtΚ ddd
8
nsObservatio
1 )( idii XXX nsobservatioPseudo
1 )(
idiiPIT UUU
tx
i
i
i xdx
dtf
1
tfti
ttK ii
d
i
i
1
1
1
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
6
1 Copulae - Examples
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gauss copula
075n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Gumbel copula
25n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from t-copula
08n 1500
00 02 04 06 08 10
00
02
04
06
08
10
Simulation from Clayton copula
25n 1500
2 Goodness-of-fit tests in one dimension
1 Estimation of the model parameter
2 Goodness-of fit testa) Crameacuter-von Mises tests
bull Fn empirical cdf
bull F cdfbull Φ weight function
Anderson-Darling
b) Critical value ndash simulation1) Simulate a sample from the copula model Cθ under H0
2) Re-estimate by ML-method
3) Calculate the test statistics
Repetition and estimation of p values
Θ θθ CCH 00 C
)()())()(( 2 xdFxxFxFnT n
7
xFxFx
1
1)(
2 Goodness-of-fit tests in more dimensions
bull Probability integral transformation (PIT) ndash mapping into the d-dimensional unit cube
~H ~C for i=1n
bull Kendallrsquos transform (K function)
Advantage one-dimensional
ndash Example Archimedean copulas
where
))(()))(()((()( 111 tUUCPtXFXFCPtΚ ddd
8
nsObservatio
1 )( idii XXX nsobservatioPseudo
1 )(
idiiPIT UUU
tx
i
i
i xdx
dtf
1
tfti
ttK ii
d
i
i
1
1
1
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
2 Goodness-of-fit tests in one dimension
1 Estimation of the model parameter
2 Goodness-of fit testa) Crameacuter-von Mises tests
bull Fn empirical cdf
bull F cdfbull Φ weight function
Anderson-Darling
b) Critical value ndash simulation1) Simulate a sample from the copula model Cθ under H0
2) Re-estimate by ML-method
3) Calculate the test statistics
Repetition and estimation of p values
Θ θθ CCH 00 C
)()())()(( 2 xdFxxFxFnT n
7
xFxFx
1
1)(
2 Goodness-of-fit tests in more dimensions
bull Probability integral transformation (PIT) ndash mapping into the d-dimensional unit cube
~H ~C for i=1n
bull Kendallrsquos transform (K function)
Advantage one-dimensional
ndash Example Archimedean copulas
where
))(()))(()((()( 111 tUUCPtXFXFCPtΚ ddd
8
nsObservatio
1 )( idii XXX nsobservatioPseudo
1 )(
idiiPIT UUU
tx
i
i
i xdx
dtf
1
tfti
ttK ii
d
i
i
1
1
1
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
2 Goodness-of-fit tests in more dimensions
bull Probability integral transformation (PIT) ndash mapping into the d-dimensional unit cube
~H ~C for i=1n
bull Kendallrsquos transform (K function)
Advantage one-dimensional
ndash Example Archimedean copulas
where
))(()))(()((()( 111 tUUCPtXFXFCPtΚ ddd
8
nsObservatio
1 )( idii XXX nsobservatioPseudo
1 )(
idiiPIT UUU
tx
i
i
i xdx
dtf
1
tfti
ttK ii
d
i
i
1
1
1
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
bull Empirical version
where
bull Kendallrsquos process
favorable asymptotic properties
bull Crameacuter-von Mises type statistic
where Φ weight function
9
2 Goodness-of-fit tests in more dimensions
))()(()( tKtKnt nnn
n
jidjdijin UUUU
nE
111
11
n
iinn ttE
ntK
1
101
1
1
0
2 )())(( dtttS nn
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
3 Serial dependence
bull Let X1 X2 Xn be univariate stationary observations EXi =μ Var(Xi )=σ2
bull If X1 X2 Xn are iid then
bull Serial dependence rarr higher variance
bull Effective sample size (ne)
where estimated variance larr bootstrap10
nX
2
)(Var
)(Var
2
e Xn
)(Var X
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
4 Bootstrap methods - Bootstrap introbull Efron (1979)
bull Let X1 X2 be iid random variables with (unknown) common distribution F ndash Xn=X1 Xn random sample
ndash Tn=tn(Xn F) random variable of interest itrsquos distribution Gn
bull Goal approximation of the distribution Gn
bull Bootstrap method ndash For given Xn we draw a simple random sample
of size m (usually m asymp n)ndash Common distribution of rsquosndash ndash Repetition 11
X X1 mmX
iX
n
iXn i
nF1
1 nmnm FtT m
X
nmG ˆ
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
4 Bootstrap methods - CBBbull Nonparametric bootstrap (sample size n)
ndash Block bootstrapbull Circular block bootstrap (CBB)
1 Let2 For some m let i1 i2 im be a uniform sample
from the set 1 2 n 3 For block size b construct nrsquo=mb (nrsquoasympn)
pseudo-data for j=1b4 Functional of interest eg bootstrap sample
mean
)(mod tt nXY
1
jijmb mYY
12
11
nn YYnY
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
4 Bootstrap methods ndash Block-length selection
DNPolitis-H White (2004) automatic block-length selection
bull Minimalize
where and
g() spectral density function
R() autocovariance function
bull Optimal block size
bull Estimation of G and D
n
bobo
n
bD
b
GMSE Xb )()( 2
2
22
k
kRkG )()0(3
4 2gD
3122
nD
Gbopt
13
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
5 Applications to wind speed maxima
bull Sample n = 2591 observations of weekly wind speed maxima for 5 German towns
bull Automatic block-length selection results
meteorologically no sense
TownOptimal block-
lengthHamburg 31Hannover 11Bremerhaven 28Fehmarn 31Schleswig 15
14
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
15
5 Applications to wind speed maxima
Method1 Fitting AR(1) modell to the data
Zt ~Extreme value distr
2 Calculation of the theoretical from AR(1) parameters
3 b optimal block size where the simulated variance of the mean first crosses the theoretical value
ttt ZXX 1
2
21
2
2
)1ˆ(
ˆ2ˆˆ2
)ˆ1(
ˆ)(Var
n
nn
nX
n
n
)(Var nX
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Bootstrap simulation results
b = 616
5 Applications to wind speed maxima
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Bootstrap simulation results
5 Applications to wind speed maxima
TownOptimal block-
lengthX-mean variance
Theoretical value
Deviation ()
IID X-mean-variance
Sample size reduction
Hamburg 8-9 00038 00034 1090 00020 185Hannover 7 00067 00071 -529 00042 159Bremerhaven 6 00073 00077 -615 00043 171Fehmarn 7 00035 00034 343 00020 174Schleswig 13 00037 00030 2279 00018 209
17
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Bremerhaven amp Fehmarn
18
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Fehmarn
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
5 Applications to wind speed maxima
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Bremerhaven amp Schleswig
19
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Bremerhaven amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
Empirical KTheoretical K
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Fehmarn amp Schleswig
20
5 Applications to wind speed maxima
GA
US
S
ST
UD
-T
GU
MB
EL
CL
AY
TO
N
00
00
00
00
04
00
00
80
00
12
Fehmarn amp Schleswig
00
00
00
00
04
00
00
80
00
12
n=1514n=2571
95 critical valueobserved statistics
00 02 04 06 08 10
00
04
08
Gumbel
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Clayton
K t
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Gauss
Empirical KTheoretical K
00 02 04 06 08 10
00
04
08
Student-t
K t
Empirical KTheoretical K
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Prediction regions (Bremerhaven amp Fehmarn)
21
5 Applications to wind speed maxima
Wind speed (ms)
Win
d sp
eed
(ms
)
0 5 10 15 20 25 30
0
5
10
15
20
25
Pred regions 50-95-998lower(5) boundsupper(95) bounds
block=1 lower boundblock=7 lower boundblock=30 lower boundblock=1 upper boundblock=7 upper boundblock=30 upper bound
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Final remarks
Conclusionsbull Copula choice is important bull Serial dependence largely influences the critical values of
GoF testsbull Block size does not have a major impact on the estimated
prediction region Future workbull Multivariate effective sample sizebull Parametric bootstrap
Acknowledgementbull We are grateful to the Doctoral School of Mathematics of
ELTE for supporting L Vargarsquos participation at SMTDA Conference
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Thank you for the attention
23
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
Referencesbull P Rakonczai A Zempleacuteni Copulas and goodness of fit tests
Recent advances in stochastic modeling and data analysis World Scientific pp 198-206 2007
bull SN Lahiri Resampling Methods for Dependent Data Springer 2003
bull DNPolitis HWhite Automatic Block-Length Selection for the Dependent Bootstrap Econometric Reviews Vol 23 pp 53-70 2004
bull P Embrechts F Lindskog A McNeil Modelling Dependence with Copulas and Applications to Risk Management Department of Mathematics ETHZ Zuumlrich 2001
bull LKish Survey Sampling J Wiley 1965
24
- Paacutel Rakonczai Laacuteszloacute Varga Andraacutes Zempleacuteni
- Outline
- 1 Copulae
- Slide 4
- Slide 5
- Slide 6
- 2 Goodness-of-fit tests in one dimension
- 2 Goodness-of-fit tests in more dimensions
- 2 Goodness-of-fit tests in more dimensions (2)
- 3 Serial dependence
- 4 Bootstrap methods - Bootstrap intro
- 4 Bootstrap methods - CBB
- 4 Bootstrap methods ndash Block-length selection
- 5 Applications to wind speed maxima
- Slide 15
- Slide 16
- 5 Applications to wind speed maxima (2)
- 5 Applications to wind speed maxima (3)
- 5 Applications to wind speed maxima (4)
- 5 Applications to wind speed maxima (5)
- 5 Applications to wind speed maxima (6)
- Final remarks
- Thank you for the attention
- References
-
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