Transcript
Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Kernel Based Estimationof Inequality Indices and Risk Measures
Arthur Charpentier
arthur.charpentier@univ-rennes1.fr
http://freakonometrics.hypotheses.org/
based on joint work with E. Flachaireinitiated by some joint work with
A. Oulidi, J.D. Fermanian, O. Scaillet, G. Geenens and D. Paindaveine
(Université de Rennes 1, 2015)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Stochastic Dominance and Related Indices
• First Order Stochastic Dominance (cf standard stochastic order, �st)
X �1 Y ⇐⇒ FX(t) ≥ FY (t),∀t⇐⇒ VaRX(u) ≤ VaRY (u),∀u
• Convex Stochastic Dominance (cf martingale property)
X �cx Y ⇐⇒ E[Y |X] = X ⇐⇒ ESX(u) ≤ ESY (u),∀u and E(X) = E(Y )
• Second Order Stochastic Dominance (cf submartingale property,stop-loss order, �icx)
X �2 Y ⇐⇒ E[Y |X] ≥ X ⇐⇒ ESX(u) ≤ ESY (u),∀u
• Lorenz Stochastic Dominance (cf dilatation order)
X �L Y ⇐⇒X
E[X] �cxX
E[Y ] ⇐⇒ LX(u) ≤ LY (u),∀u
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Stochastic Dominance and Related Indices
• Parametric Model(s)E.g. N (µX , σ2
X) �1 N (µY , σ2Y )⇐⇒ µX ≤ µY and σ2
X = σ2Y
E.g. N (µX , σ2X) �cx N (µY , σ2
Y )⇐⇒ µX = µY and σ2X ≤ σ2
Y
E.g. N (µX , σ2X) �2 N (µY , σ2
Y )⇐⇒ µX ≤ µY and σ2X ≤ σ2
Y
Or other parametric distribution. E.g. a lognormal distribution for losses
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Stochastic Dominance and Related Indices
• Non-parametric Model(s)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Nonparametric estimation of the density
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Agenda
sample {y1, · · · , yn} −→ fn(·)↗↘
Fn(·) or F−1n (·)
R(fn)
• Estimating densities of copulas◦ Beta kernels◦ Transformed kernels• Combining transformed and Beta kernels• Moving around the Beta distribution◦ Mixtures of Beta distributions◦ Bernstein Polynomials• Some probit type transformations
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Non parametric estimation of copula densitysee C., Fermanian & Scaillet (2005), bias of kernel estimators at endpoints
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Kernel based estimation of the uniform density on [0,1]
Dens
ity
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40.
60.
81.
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Kernel based estimation of the uniform density on [0,1]
Dens
ity
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Non parametric estimation of copula density
e.g. E(c(0, 0, h)) = 14 · c(u, v)− 1
2 [c1(0, 0) + c2(0, 0)]∫ 1
0ωK(ω)dω · h+ o(h)
0.2 0.4 0.6 0.8
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Estimation of Frank copula
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Estimation of Frank copula
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with a symmetric kernel (here a Gaussian kernel).
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Non parametric estimation of copula density
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01
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Standard Gaussian kernel estimator, n=100
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
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Standard Gaussian kernel estimator, n=1000
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
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Standard Gaussian kernel estimator, n=10000
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
Nice asymptotic properties, see Fermanian et al. (2005)... but still: on finitesample, bad behavior on borders.
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Beta kernel idea (for copulas)see Chen (1999, 2000), Bouezmarni & Rolin (2003),
Kxi(u) ∝ exp(− (u− xi)2
h2
)vs. Kxi(u) ∝
(u
x1,ib
1 [1− u1]x1,i
b
)·(u
x2,ib
2 [1− u2]x2,i
b
)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Beta kernel idea (for copulas)Beta (independent) bivariate kernel , x=0.0, y=0.0 Beta (independent) bivariate kernel , x=0.2, y=0.0 Beta (independent) bivariate kernel , x=0.5, y=0.0
Beta (independent) bivariate kernel , x=0.0, y=0.2 Beta (independent) bivariate kernel , x=0.2, y=0.2 Beta (independent) bivariate kernel , x=0.5, y=0.2
Beta (independent) bivariate kernel , x=0.0, y=0.5 Beta (independent) bivariate kernel , x=0.2, y=0.5 Beta (independent) bivariate kernel , x=0.5, y=0.5
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Beta kernel idea (for copulas)
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Estimation of Frank copula
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0.60.8
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Estimation of the copula density (Beta kernel, b=0.1)
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0.60.8
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Estimation of the copula density (Beta kernel, b=0.05)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Beta kernel idea (for copulas)
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01
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Beta kernel estimator, b=0.05, n=100
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
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01
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Beta kernel estimator, b=0.02, n=1000
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
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01
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Beta kernel estimator, b=0.005, n=10000
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Transformed kernel idea (for copulas)[0, 1]× [0, 1]→ R× R→ [0, 1]× [0, 1]
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Transformed kernel idea (for copulas)
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Estimation of Frank copula
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Estimation of Frank copula
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Estimation of Frank copula
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Transformed kernel idea (for copulas)
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01
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Transformed kernel estimator (Gaussian), n=100
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
0.0 0.2 0.4 0.6 0.8 1.0
01
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Transformed kernel estimator (Gaussian), n=1000
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
0.0 0.2 0.4 0.6 0.8 1.0
01
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Transformed kernel estimator (Gaussian), n=10000
Estimation of the density on the diagonal
Dens
ity o
f the
est
imat
or
see Geenens, C. & Paindaveine (2014) for more details on probit transformationfor copulas.
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Combining the two approachesSee Devroye & Györfi (1985), and Devroye & Lugosi (2001)
... use the transformed kernel the other way, R→ [0, 1]→ R
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Devroye & Györfi (1985) - Devroye & Lugosi (2001)Interesting point, the optimal T should be F ,
thus, T can be Fθ
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Heavy Tailed distributionLet X denote a (heavy-tailed) random variable with tail index α ∈ (0,∞), i.e.
P(X > x) = x−αL1(x)
where L1 is some regularly varying function.
Let T denote a R→ [0, 1] function, such that 1− T is regularly varying atinfinity, with tail index β ∈ (0,∞).
Define Q(x) = T−1(1− x−1) the associated tail quantile function, thenQ(x) = x1/βL?2(1/x), where L?2 is some regularly varying function (the de Bruynconjugate of the regular variation function associated with T ). Assume here thatQ(x) = bx1/β
Let U = T (X). Then, as u→ 1
P(U > u) ∼ (1− u)α/β .
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Heavy Tailed distributionsee C. & Oulidi (2007), α = 0.75−1 , T0.75−1 , T0.65−1︸ ︷︷ ︸
lighter
, T0.85−1︸ ︷︷ ︸heavier
and Tα
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Heavy Tailed distributionsee C. & Oulidi (2007), impact on quantile estimation ?
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Heavy Tailed distributionsee C. & Oulidi (2007), impact on quantile estimation ? bias ? m.s.e. ?
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Which transformation ?
GB2 : t(y; a, b, p, q) = |a|yap−1
bapB(p, q)[1 + (y/b)a]p+q , for y > 0,
GB2q→∞
��a=1
��p=1
##
q=1
((GG
a→0
��a=1
��
p=1
��
Beta2
q→∞ww
SM
q→∞xx
q=1''
Dagum
p=1
��Lognormal Gamma Weibull Champernowne
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Estimating a density on R+
• Stay on R+ : xi’s• Get on [0, 1] : ui = T
θ(xi) (distribution as uniform as possible)
◦ Use Beta Kernels on ui’s◦ Mixtures of Beta distributions on ui’s◦ Bernstein Polynomials on ui’s• Get on R : use standard kernels (e.g. Gaussian)◦ On x?i = log(xi)◦ On x?i = BoxCox
λ(xi)
◦ On x?i = Φ−1[Tθ(xi)]
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Beta kernel
g(u) =n∑i=1
1n· b(u; Ui
h,
1− Uih
)u ∈ [0, 1].
with some possible boundary correction, as suggested in Chen (1999),u
h→ ρ(u, h) = 2h2 + 2.5− (4h4 + 6h2 + 2.25− u2 − u/h)1/2
Problem : choice of the bandwidth h? ? Standard loss function
L(h) =∫
[gn(u)− g(u)]2du =∫
[gn(u)]2du− 2∫gn(u) · g(u)du︸ ︷︷ ︸
CV (h)
+∫
[g(u)]2du
where
CV (h) =(∫
gn(u)du)2− 2n
n∑i=1
g(−i)(Ui)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Mixture of Beta distributions
g(u) =k∑j=1
πj · b(u; αj , βj
)u ∈ [0, 1].
Problem : choice the number of components k (and estimation...). Use ofstochastic EM algorithm (or sort of) see Celeux & Diebolt (1985).
Bernstein approximation
g(u) =m∑k=1
[mωk] · b (u; k,m− k) u ∈ [0, 1].
where ωk = G
(k
m
)− G
(k − 1m
).
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
On the log-transformWith a standard Gaussian kernel
fX(x) = 1n
n∑i=1
φ(x;xi, h)
A Gaussian kernel on a log transform,
fX(x) = 1xfX?(log x) = 1
n
n∑i=1
λ(x; log xi, h)
where λ(·;µ, σ) is the density of the log-normal distribution. Here, in 0,
bias[fX(x)] ∼ h2
2 [fX(x) + 3x · f ′X(x) + x2 · f ′′X(x)]
andVar[fX(x)] ∼ fX(x)
xnh
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
On the Box-Cox-transformMore generally, instead of transformed sample Yi = log[Xi], consider
Yi = Xλi − 1λ
when λ 6= 0.
Find the optimal transformation using standard regression techniques (leastsquares)
X?i = Xλ?
i − 1λ?
when λ? 6= 0
and X?i = log[Xi] if λ? = 0. The density estimation is here
fX(x) = xλ?−1fX?
(xλ
? − 1λ?
)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Illustration with Log-normal samplesStandard kernel (− Silvermans’s rule h?)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Illustration with Log-normal samplesLog transform, x?i = log xi + Gaussian kernel
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Illustration with Log-normal samplesProbit-type transform, x?i = Φ−1[T
θ(xi)] + Gaussian kernel
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Illustration with Log-normal samplesBox-Cox transform, x?i = BoxCox
λ(xi) + Gaussian kernel
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Illustration with Log-normal samplesui = T
θ(xi) + Mixture of Beta distributionsl
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Illustration with Log-normal samplesui = T
θ(xi) + Beta kernel estimation
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Quantities of interestStandard statistical quantities
• miae,(∫ ∞
0
∣∣∣fn(x)− f(x)∣∣∣ dx
)
• mise,(∫ ∞
0
[fn(x)− f(x)
]2dx)
• miaew,(∫ ∞
0
∣∣∣fn(x)− f(x)∣∣∣ |x|dx)
• misew,(∫ ∞
0
[fn(x)− f(x)
]2x2 dx
)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Quantities of interest
Inequality indices and risk measures, based on F (x) =∫ x
0f(t)dt,
• Gini, 1µ
∫ ∞0
F (t)[1− F (t)]dt
• Theil,∫ ∞
0
t
µlog(t
µ
)f(t)dt
• Entropy −∫ ∞
0f(t) log[f(t)]dt
• VaR-quantile, x such that F (x) = P(X ≤ x) = α, i.e. F−1(α)
• TVaR-expected shorfall, E[X|X > F−1(α)]
where µ =∫ ∞
0[1− F (x)]dx.
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Computations AspectsHere, for each method, we return two functions,
• function fn(·)
• a random generator for distribution fn(·)
◦ H-transform and Gaussian kernel
draw i ∈ {1, · · · , n} and X = H−1(Z) where Z ∼ N (H(xi), b2)
◦ H-transform and Beta kernel
draw i ∈ {1, · · · , n} and X = H−1(U) where U ∼ B(H(xi)/h, [1−H(xi)]/h)
◦ H-transform and Beta mixture
draw k ∈ {1, · · · ,K} and X = H−1(U) where U ∼ B(αk,βk)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
◦ ‘standard’ Gaussian kernel (benchmark)
draw i ∈ {1, · · · , n}, and X ∼ N (xi, b2) (almost)
up to some normalization, · 7→ fn(·)∫∞0 fn(x)dx
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
MISE
∫ ∞0
[f (s)n (x)− f(x)
]2dx
●
Singh−Maddala
MISE
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● ●●● ●● ●●● ●● ●
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● ●●● ●● ●●● ●● ●
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standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
0.00
0
0.00
5
0.01
0
0.01
5
●
Mixed Singh−Maddala
MISE
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● ●● ●● ●●● ●● ●●●● ●
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●●●
standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
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0.15
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
MIAE
∫ ∞0|f (s)n (x)− f(x)|dx
●
Singh−Maddala
MIAE
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standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
0.00
0.05
0.10
0.15
0.20
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Mixed Singh−Maddala
MIAE
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standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
0.05
0.10
0.15
0.20
0.25
0.30
40
Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Gini Index
1µ
(s)n
∫ ∞0
F (s)n (t)[1− F (s)
n (t)]dt
Singh−Maddala
Gini Index
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●
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standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
0.45
0.50
0.55
0.60
Mixed Singh−Maddala
Gini Index
●
●●
●
●
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●●
standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
0.55
0.60
0.65
0.70
41
Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Value-at-Risk, 95%
Q(s)n (α) = inf{x, α ≤ F (s)
n (x)}
●
Singh−Maddala
Quantile 95%
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standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
10 20 30 40 50
Mixed Singh−Maddala
Quantile 95%
●●
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●
●
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●
standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
2.0
2.5
3.0
3.5
4.0
4.5
42
Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Value-at-Risk, 99%
Q(s)n (α) = inf{x, α ≤ F (s)
n (x)}
Singh−Maddala
Quantile 99%
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standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
6 8 10 12 14 16 18
●
Mixed Singh−Maddala
Quantile 99%
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●● ●
standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
2 4 6 8 10
43
Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Tail Value-at-Risk, 95%
E[X|X > Q(s)n (α)]
Singh−Maddala
Expected Shortfall 95%
●
●
●●
●
●●
●
standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
4.0
4.5
5.0
5.5
6.0
6.5
●
Mixed Singh−Maddala
Expected Shortfall 95%
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●
standard kernellog kernel
log mixboxcox kernel
boxcox mixprobit kernelbeta kernel
beta mix
4 6 8 10 12
44
Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Possible conclusion ?
• estimating densities on transformated data is definitively a good idea
• but we need to find a good transformationX parametric + betaX parametric + probitX log-transformX Box-Cox
0.719
0.719
0.7190.719
0.719
0.719
0.719
0.7191.428
2.137
47.75
48.00
48.25
48.50
48.75
−4.8 −4.4 −4.0 −3.6longitude
latit
ude
0.0110.7191.4282.1372.845
0.719 0.719
0.7190.719
0.719
0.719
0.7190.719
0.719
1.428
1.428
1.428
1.428
1.428
1.4281.428
1.428
1.428
1.428
1.428
1.428
1.428
1.428
1.428
1.428
2.137
2.137
47.75
48.00
48.25
48.50
48.75
−4.8 −4.4 −4.0 −3.6longitude
latit
ude
0.0110.7191.4282.1372.845
(joint work with E. Gallic)
45
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