Chapter V Extreme Value Theory and Frequency Analysis 5.1 INTRODUCTION TO EXTREME VALUE THEORY Most statistical methods are concerned primarily with what goes on in the center of a statistical distribution, and do not pay particular attention to the tails of a distribution, or in other words, the most extreme values at either the high or low end. Extreme event risk is present in all areas of risk management – market, credit, day to day operation, and insurance. One of the greatest challenges to a risk manager is to implement risk management tools which allow for modeling rare but damaging events, and permit the measurement of their consequences. Extreme value theory (EVT) plays a vital role in these activities. The standard mathematical approach to modeling risks uses the language of probability theory. Risks are random variables, mapping unforeseen future states of the world into values representing profits and losses. These risks may be considered individually, or seen as part of a stochastic process where present risks depend on previous risks. The potential values of a risk have a probability distribution which we will never observe exactly although past losses due to similar risks, where available, may provide partial information about that distribution. Extreme events occur when a risk takes values from the tail of its distribution. We develop a model for risk by selecting a particular probability distribution. We may have estimated this distribution through statistical analysis of empirical data. In this case EVT is a tool which attempts to provide us with the best possible estimate of the tail area of the distribution. However, even in the absence of useful historical data, EVT provides guidance on the kind of distribution we should select so that extreme risks are handled conservatively. There are two principal kinds of model for extreme values. The oldest group of models is the block maxima models; these are models for the largest observations collected from large samples of identically distributed observations. For example, if we record daily or hourly losses and profits from trading a particular instrument or
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Chapter V
Extreme Value Theory and Frequency Analysis
5.1 INTRODUCTION TO EXTREME VALUE THEORY
Most statistical methods are concerned primarily with what goes on in the center of a
statistical distribution, and do not pay particular attention to the tails of a distribution,
or in other words, the most extreme values at either the high or low end. Extreme
event risk is present in all areas of risk management – market, credit, day to day
operation, and insurance. One of the greatest challenges to a risk manager is to
implement risk management tools which allow for modeling rare but damaging
events, and permit the measurement of their consequences. Extreme value theory
(EVT) plays a vital role in these activities.
The standard mathematical approach to modeling risks uses the language of
probability theory. Risks are random variables, mapping unforeseen future states of
the world into values representing profits and losses. These risks may be considered
individually, or seen as part of a stochastic process where present risks depend on
previous risks. The potential values of a risk have a probability distribution which we
will never observe exactly although past losses due to similar risks, where available,
may provide partial information about that distribution. Extreme events occur when a
risk takes values from the tail of its distribution.
We develop a model for risk by selecting a particular probability distribution. We may
have estimated this distribution through statistical analysis of empirical data. In this
case EVT is a tool which attempts to provide us with the best possible estimate of the
tail area of the distribution. However, even in the absence of useful historical data,
EVT provides guidance on the kind of distribution we should select so that extreme
risks are handled conservatively.
There are two principal kinds of model for extreme values. The oldest group of
models is the block maxima models; these are models for the largest observations
collected from large samples of identically distributed observations. For example, if
we record daily or hourly losses and profits from trading a particular instrument or
Chapter V: Extreme Value Theory and Frequency Analysis
group of instruments, the block maxima/minima method provides a model which may
be appropriate for the quarterly or annual maximum of such values. The block
maxima/minima methods are fitted with the generalized extreme value (GEV)
distribution.
A more modern group of models is the peaks-over-threshold (POT) models; these are
models for all large observations which exceed a high threshold. The POT models are
generally considered to be the most useful for practical applications, due to a number
of reasons. First, by taking all exceedances over a suitably high threshold into
account, they use the data more efficiently. Second, they are easily extended to
situations where one wants to study how the extreme levels of a variable Y depend on
some other variable X – for instance, Y may be the level of tropospheric ozone on a
particular day and X a vector of meteorological variables for that day. This kind of
problem is almost impossible to handle through the annual maximum method. The
POT methods are fitted with the generalized Pareto distribution (GPD).
5.2 GENERALIZED EXTREME VALUE DISTRIBUTION
The role of the generalized extreme value (GEV) distribution in the theory of
extremes is analogous to that of the normal distribution in the central limit theory for
sums of random variables. The normal distribution proves to be the important limiting
distribution for sample sums or averages, as is made explicit in the central limit
theorem, the GEV distribution also proves to be important in the study of the limiting
behavior of sample extrema. The three-parameter distribution function of the standard
GEV is given by:
⎪⎩
⎪⎨
⎧
=−
≠−
+−=
−−
−
,0)exp(
,0))1(exp()(
/1
,,
ξ
ξσµξ
σµ
ξ
σµξ
ife
ifx
xHx
(5.2.1)
where 01 >−
+σµξ x is such that 01 >+ xξ . µ and 0>σ are known as the location
and scale parameters, respectively. ξ is the all-important shape parameter which
determines the nature of the tail distribution. The extreme value distribution in
equation (5.2.1) is generalized in the sense that the parametric form subsumes three
types of distributions which are known by other names according to the value of ξ :
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Chapter V: Extreme Value Theory and Frequency Analysis
when 0>ξ we have the Frechet distribution with ξα /1= ; when 0<ξ we have the
Weibull distribution with shape ξα /1−= ; when 0=ξ we have the Gumbel
distribution. The Weibull is a short-tailed distribution with a so-called finite right end
point. The Gumbel and the Frechet have infinite right end points but the decay of tail
of the Frechet is much slower than that of the Gumbel.
Here are a few basic properties of the GEV distribution. The mean exists if 1<ξ and
the variance if 21
<ξ ; more generally, the 'th moment exists if kk1
<ξ . The mean
and variance are given by
}1)1({)(1 −−Γ+== ξξσµXEm , )1( <ξ
)}1()21({}){( 22
222
12 ξξξσ
−Γ−−Γ=−= mXEm , )2/1( <ξ
One objection to the extreme value distributions is that many processes rarely produce
observations that are independent and identically distributed. However, there is an
extensive theory of extreme value theory for non-IID processes. A second objection is
that sometimes it is argued that alternative distributional families fit the data better –
for example, in the 1970s there was a lengthy debate among hydrologists over the use
of extreme value distributions as compared with those of log Pearson type III. There
is no universal solution to this kind of debate.
5.2.1 The Fisher-Tippett Theorem
The Fisher-Tippett theorem is the fundamental result in EVT and can be considered to
have the same status in EVT as the central limit theorem has in the study of sums. The
theorem describes the limiting behavior of appropriately normalized sample maxima.
Suppose are a sequence of independent random variables with a common
distribution function F; in other words
K,, 21 XX
)()( xXPxF j ≤= for each and . The
distribution function of the maximum of the first observations
is given by the n 'th power of F :
j x
n
),,max( 1 nn XXM K=
{ } { }xXxXxXPxMP nn ≤≤≤=≤ ,,, 21 L
104
Chapter V: Extreme Value Theory and Frequency Analysis
{ } { } { }xXPxXPxXP n ≤≤≤= L21
)(xF n=
Suppose further that we can find sequences of real numbers and such
that , the sequence of normalized maxima, converge in distribution. That
is
0>na nb
nnn abM /)( −
( ){ } { }nnnnnn bxaMPxabMP +≤=≤− /
)( nnn bxaF +=
as )(xH→ ∞→n (5.2.2)
for some non-degenerate distribution function . If this condition holds we say
that is in the maximum domain of attraction of
)(xH
F H and we write . It
was shown by Fisher and Tippett (1928) that
)(HMDAF ∈
ξξ somefor type theof is )( HHHMDAF ⇒∈
Thus, if we know that suitably normalized maxima converge in distribution, then the
limit distribution must be an extreme value distribution for some value of the
parameters ξ , µ , and σ .
The class of distributions for which the condition (5.2.2) holds is large. Gnedenko
(1943) showed that for
F
0>ξ , )( ξHMDAF ∈ if and only if , for
some slowly varying function L(x). This result essentially says that if the tail of the
distribution function decays like a power function, then the distribution is in the
domain of attraction of the Frechet. The class of distributions where the tail decays
like a power function is quite large and includes the Pareto, Burr, loggamma, Cauchy
and t-distributions as well as various mixture models. These are all so-called heavy
tailed distributions. Distributions in the maximum domain of attraction of the Gumbel
include the normal, exponential, gamma and lognormal distributions. The
lognormal distribution has a moderately heavy tail and has historically been a popular
model for loss severity distributions; however it is not as heavy-tailed as the
distributions in for
)()(1 /1 xLxxF ξ−=−
)(xF
)( 0HMDA
)( ξHMDA 0<ξ . Distributions in the domain of attraction of the
Weibull ( forξH 0<ξ ) are short tailed distributions such as the uniform and beta
distributions.
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Chapter V: Extreme Value Theory and Frequency Analysis
The Fisher-Tippett theorem suggests the fitting of the GEV to data on sample
maxima, when such data can be collected. There is much literature on this topic
particularly in hydrology where the so-called annual maxima method has a long
history.
5.2.2 Fitting the GEV Distribution
The GEV distribution can be fitted using various methods – probability weighted
moments, maximum likelihood, Bayesian methods. The latter two require numerical
computation, and one disadvantage is that the computations are not easily performed
with standard statistical packages. However, there are some programs available which
are specifically tailored to extreme values. In implementing maximum likelihood, it is
assumed that the block size is quite large so that, regardless of whether the underlying
data are dependent or not, the block maxima observations can be taken to be
independent.
For the GEV, the density ),,;( ξσµxp is obtained by differentiating (5.2.1) w.r.t. .
The likelihood based on observations is
x
nYYY ,,, 21 L
∏=
N
iiYp
1
),,;( ξσµ
and so the log likelihood is given by
( )ξ
σµ
ξσµ
ξξ
σξσµ/1
11log11log,,−
∑∑ ⎟⎠⎞
⎜⎝⎛ −+−⎟
⎠⎞
⎜⎝⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−−=
i
i
i
iY
YYNl (5.2.3)
which must be maximized subject to the parameter constraints that 0>σ and
01 >−
+σµ
ξ iY for all i . While this represents an irregular likelihood problem, due to
the dependence of the parameter space on the values of the data, the consistency and
asymptotic efficiency of the resulting MLEs can be established for the case when
21
−>ξ .
In determining the number and size of blocks ( and respectively) a trade-off
necessarily takes place: roughly speaking, a large value of m leads to a more accurate
n m
106
Chapter V: Extreme Value Theory and Frequency Analysis
approximation of the block maxima distribution by a GEV distribution and a low bias
in the parameter estimates; a large value of gives more block maxima data for the
ML estimation and leads to a low variance in the parameter estimates. Notes also that,
in the case of dependent data, somewhat larger block sizes than are used in the IID
case may be advisable; dependence generally has the effect that convergence to the
GEV distribution is slower, since the effective sample size is
n
θm , which is smaller
than . m
The following exploratory analyses can be done while fitting a GEV or GPD model to
a set of given data:
Time series plot
Autocorrelation plot
Histogram on the log scale
Quantile-Quantile plot against the exponential distribution
Gumbel plot (Mondal, 2006d, p. 338)
Sample mean excess plot (also called the mean residual life plot in survival
data)
Plot of empirical distribution function
Quantile-Quantile plot of residuals of a fitted model
Example 5.2.1 The highest water level during the period from 1 March to 15
May of different years at the Sunamganj station of the Surma river in Bangladesh is