Consider the stochastic behavior of the maximum
of
identically distributed random variables
with cumulative distribution function (cdf)
. From a risk management perspective
is the
negative return at day
. The cdf of
is
![$\displaystyle {\P}(M_n \le x) = {\P}(X_1 \le x, \ldots, X_n \le x) = \prod_{t=1}^n {\P}(X_t \le x) = F^ n (x).$](sfehtmlimg3403.gif) |
(18.1) |
We are only considering unbounded random variables
, i.e.
for all
Obviously it holds that
for all
, when
, and thus
. The maximum of
unbounded random variables increases over all boundaries. In order
to achieve a non-degenerate behavior limit,
has to be
standardized in a suitable fashion.
Definition 18.1 (Maximum Domain of Attraction)
The random variable
![$ X_t$](sfehtmlimg506.gif)
belongs to the
maximum domain of
attraction (MDA) of a non-degenerate distribution
![$ G$](sfehtmlimg59.gif)
, if for
suitable sequences
![$ c_n
> 0, d_n $](sfehtmlimg3409.gif)
it holds that:
![$\displaystyle \frac{M_n - d_n}{c_n} \stackrel{{\cal L}}{\longrightarrow} G$](sfehtmlimg3410.gif)
for
i.e.
![$ F^ n (c_n x + d_n) \to G(x)$](sfehtmlimg3412.gif)
at all continuity points
![$ x$](sfehtmlimg40.gif)
of the cdf
![$ G(x)$](sfehtmlimg3413.gif)
.
It turns out that only a few distributions
can be considered
as the asymptotic limit distribution of the standardized maximum
. They are referred to as the extreme value
distriubtions. These are the
following three distribution functions:
Fréchet: |
for
![$ \alpha > 0 ,$](sfehtmlimg3415.gif) |
|
|
Gumbel: |
![$ G_0 (x) = \exp \{ - e^ {-x}\},\ x \in
\mathbb{R} ,$](sfehtmlimg3416.gif) |
|
|
Weibull: |
for
![$ \alpha < 0 .$](sfehtmlimg3418.gif) |
Fig.:
Fréchet (red), Gumbel (black) and Weibull distributions (blue).
SFEevt1.xpl
|
The Fréchet distributions are concentrated on the
non-negative real numbers
, while the Weibull
distribution, on the other hand, on
whereas the
Gumbel distributed random variables can attain any real number.
Figure 17.1 displays the density function of the Gumbel
distribution, the Fréchet distribution with parameter
and the Weibull distribution with parameter
. All three distributions types can be displayed in a
single Mises form:
Definition 18.2 (Extreme Value Distributions)
The
generalized extreme value distribution (GEV =
generalized extreme value) with the
form parameter
![$ \gamma
\in
\mathbb{R} $](sfehtmlimg3423.gif)
has the distribution function:
is the Gumbel distribution,
whereas
is linked to the Fréchet- and Weibull
distributions by the following
relationships:
![$\displaystyle G_\gamma (\frac{x-1}{\gamma}) = G _{1,1/\gamma} (x)\ $](sfehtmlimg3429.gif)
for
![$\displaystyle G_\gamma (- \frac{x+1}{\gamma} ) = G_{2,- 1/\gamma} (x)$](sfehtmlimg3431.gif)
for
This definition describes the standard
form of the GEV distributions. In general
we can change the center and the scale to obtain other GEV
distributions:
with the
form parameter
, the location parameter
and the scale parameter
For asymptotic theory this does not matter since the
standardized sequences
can be always chosen so that
the asymptotic distribution
has the standard form (
). An important result of the asymptotic distribution
of the maximum
is the Fisher-Tippett
theorem:
Theorem 18.1
If there exists sequences
![$ c_n
> 0, d_n $](sfehtmlimg3409.gif)
and a non-degenerate
distribution
![$ G$](sfehtmlimg59.gif)
, so that
![$\displaystyle \frac{M_n - d_n}{c_n} \stackrel{{\cal L}}{\longrightarrow} G$](sfehtmlimg3410.gif)
for
then
![$ G$](sfehtmlimg59.gif)
is a GEV distribution.
Proof:
As a form of clarification the basic ideas used to prove this
central result are outlined. Let
, and
represent
the integer part of
. Since
is the distribution
function of
, due to our assumptions on the asymptotic
distribution of
it holds that
On the other hand
it also holds that
![$\displaystyle F^ {[nt]} (c_n x + d_n) = \{ F^ n (c_n x + d_n) \} ^ {\frac{[nt]}{n}}
\longrightarrow G^ t (x)\ $](sfehtmlimg3445.gif)
for
In
other words this means that
for
. According to the Lemma, which is stated below,
this is only possible when
and
![$\displaystyle G^ t(x) = G(b(t) x + a(t) ) ,\ t > 0,\ x \in \mathbb{R}.$](sfehtmlimg3448.gif) |
(18.2) |
This relationship holds for arbitrary values
. We use it in
particular for arbitrary
and
and obtain
![$\displaystyle b (st) =b (s)\ b (t),\ a (st) = b (t) a (s)+ a (t).$](sfehtmlimg3451.gif) |
(18.3) |
The functional equations (17.2), (17.3) for
have only one solution, when
is one of the
distributions
or
, that is,
must be a GEV distribution.
Lemma 18.1 (Convergence Type Theorem)
Let
![$ U_1, U_2, \ldots, V, W$](sfehtmlimg3455.gif)
be random variables,
![$ b_n, \beta _n > 0,\
a_n, \alpha_n \in \mathbb{R}.$](sfehtmlimg3456.gif)
If
in distribution for
![$ n\to \infty$](sfehtmlimg3407.gif)
, then it holds that:
![$\displaystyle \frac{U_n - \alpha_n}{\beta_n} \stackrel{{\cal L}}{\longrightarrow} W\ $](sfehtmlimg3458.gif)
if and only if
In this case
![$ W$](sfehtmlimg1900.gif)
has the same distribution as
![$ bV+a$](sfehtmlimg3460.gif)
.
Notice that the GEV distributions are identical to the so called
max-stable distributions, by
which for all
the maximum
of
i.i.d. random
variables
have the same distribution as
for appropriately chosen
.
Fig.:
PP plot for the normal distribution and pseudo random variables with extreme value distributions. Fr'echet (upper left), Weibull (upper right) and Gumbel (below).
SFEevt2.xpl
|
Figure 17.2 shows the so called normal plot, i.e., it
compares the graph of the cdf of the normal distribution with the
one in Section 17.2 for the special case
with computer generated random variables that have
a Gumbel distribution, Fréchet distribution with parameter
and a Weibull distribution with parameter
respectively. The differences with the normally distributed
random variables, which would have approximately a straight line
in a normal plot, can be clearly seen.
If the maximum of i.i.d. random variables converges in
distribution after being appropriately standardized, then the
question arises which of the three GEV distributions is the
asymptotic distribution. The deciding factor is how fast the
probability for extremely large observations decreases beyond a
threshold
, when
increases. Since this exceedance
probability plays an important role
in extreme value theory, we will introduce some more notations:
The relationship between the exceedance probability
and the distribution of the maxima
will
become clear with the following theorem.
Theorem 18.2
a) For
![$ 0 \le \tau \le \infty$](sfehtmlimg3469.gif)
and every sequence of real numbers
![$ u_n, n \ge 1,$](sfehtmlimg3470.gif)
it holds for
![$ n\to \infty$](sfehtmlimg3407.gif)
that
![$\displaystyle n\overline{F} (u_n) \to \tau \, \; \;$](sfehtmlimg3471.gif)
if and only if
b)
![$ F$](sfehtmlimg58.gif)
belongs to the maximum domain
of attraction of the GEV distribution
![$ G$](sfehtmlimg59.gif)
with the standardized
sequences
![$ c_n, d_n$](sfehtmlimg3436.gif)
exactly when
![$\displaystyle n \overline{F} (c_n x + d_n) \to - \log G(x)$](sfehtmlimg3473.gif)
for all
The exceedance probability of the Fréchet distribution
behaves like
for
,
because the exponential function around 0 is approximately linear,
i.e.,
![$\displaystyle \overline{G}_{1,\alpha}(x) = \frac{1}{x^ \alpha} \{1 + {\scriptstyle \mathcal{O}}(1) \}$](sfehtmlimg3478.gif)
for
Essentially all of the distributions that belong to the MDA of
this Fréchet distribution show the same behavior;
is almost constant for
, or more
specifically: a slowly varying function.
Definition 18.3
A positive measurable function
![$ L$](sfehtmlimg999.gif)
in
![$ (0,\infty)$](sfehtmlimg3359.gif)
is called
slowly varying, if for all
![$\displaystyle \frac{L(tx)}{L(x)} \to 1$](sfehtmlimg3481.gif)
for
Typical slowly varying functions are, in addition to constants,
logarithmic growth rates, for example
.
Theorem 18.3
![$ F$](sfehtmlimg58.gif)
belongs to the maximum domain of attraction of the Fréchet
distribution
![$ G_{1,\alpha}$](sfehtmlimg3475.gif)
for some
![$ \alpha>0$](sfehtmlimg2047.gif)
, if and only if
![$ x^ \alpha \overline{F}(x) = L(x)$](sfehtmlimg3483.gif)
is a slowly varying
function. The random variables
![$ X_t$](sfehtmlimg506.gif)
with the distribution
function
![$ F$](sfehtmlimg58.gif)
are unbounded, i.e.,
![$ F(x) < 1$](sfehtmlimg3404.gif)
for all
![$ x < \infty,$](sfehtmlimg3484.gif)
and it holds that
with
![$ c_n = F^{-1}(1 - \frac{1}{n}).$](sfehtmlimg3486.gif)
For the description of the standardized sequence
we have
used the following notation.
is an extreme quantile of the
distribution
, and it holds that
Definition 18.4 (Quantile Function)
If
![$ F$](sfehtmlimg58.gif)
is a distribution function, we call the generalized inverse
the
quantile function. It then
holds that
![$ {\P}( X_1 \le F^{-1}(\gamma ) )=\gamma $](sfehtmlimg3490.gif)
, i.e.,
![$ F^{-1}(\gamma )$](sfehtmlimg3491.gif)
is the
![$ \gamma$](sfehtmlimg1613.gif)
-
quantile
of the distribution
![$ F$](sfehtmlimg58.gif)
.
If
is strictly monotonic increasing and continuous, then
is the generalized inverse of
.
There is a corresponding criterion for the Weibull distribution
that can be shown using the relationship
. Random variables, whose maxima are
asymptotically Weibull distributed, are by all means bounded,
i.e., there exists a constant
, such that
with probability 1. Therefore, in financial applications they are
only interesting in special situations where using a type of
hedging strategy, the loss, which can result from an investment,
is limited. In order to prohibit continuous differentiations in
various cases, in the following we will mainly discuss the case
where the losses are unbounded. The cases in which losses are
limited can be dealt with in a similar fashion.
Fréchet distributions appear as asymptotic distributions of the
maxima of those random variables whose probability of values
beyond
only slowly decreases with
, whereas only bounded
random variables belong to the maximum domain of attraction of
Weibull distributions. Many known distributions such as the
exponential or the normal distribution do not belong to either one
of the groups. It is likely that in such cases the distribution of
the appropriate standardized maxima converges to a Gumbel
distribution. The general conditions need for this are however
more complicated and more difficult to prove than they were for
the Fréchet distribution.
Theorem 18.4
The distribution function
![$ F$](sfehtmlimg58.gif)
of the unbounded random variables
![$ X_t$](sfehtmlimg506.gif)
belongs to the maximum domain of attraction of the Gumbel
distribution if measurable scaling functions
![$ c(x), g(x)
> 0$](sfehtmlimg3496.gif)
as well as an absolute continuous function
![$ e(x) > 0$](sfehtmlimg3497.gif)
exist with
![$ c(x) \to c > 0, \ g(x)
\to 1, e'(x) \to 0$](sfehtmlimg3498.gif)
for
![$ x \to \infty$](sfehtmlimg3477.gif)
so that for
In this case it holds that
with
![$ d_n = F^{-1}(1- \frac{1}{n})$](sfehtmlimg3502.gif)
and
![$ c_n = e(d_n).$](sfehtmlimg3503.gif)
As a function
, the average excess
function can be used:
which will be considered in more detail in the following.
The exponential distribution with parameter
has the
distribution function
so that
fulfills the
conditions stipulated in the theorem with
and
. The maximum
of
independent exponentially distributed random variables with
parameter
thus converges in distribution to the Gumbel
distribution:
![$\displaystyle \lambda (M_n - \frac{1}{\lambda} \ \log \ n) \stackrel{{\cal L}}{\longrightarrow} G_0$](sfehtmlimg3510.gif)
for
In general, however, the conditions are not so easy to check.
There are other simple sufficient conditions with which it can be
shown, for example, that also the normal distribution belongs to
the maximum domain of attraction of the Gumbel distribution. If,
for example,
is the maximum of
independent standard
normally distributed random variables, then it holds that
SFEevtex1.xpl
Another member of the distributions in the maximum domain of
attraction of the Fréchet distribution
is the
Pareto distribution with the
distribution function
as well as all other distributions with Pareto
tails, i.e., with
![$\displaystyle \overline{F}(x) = \frac{\kappa}{x^ \alpha} \{ 1 + {\scriptstyle \mathcal{O}}(1) \} \; \; \;$](sfehtmlimg3516.gif)
for
Since
for
behaves
here like
,
for
is identical to
, and
![$\displaystyle \frac{M_n}{(\kappa n)^ {1/\alpha}} \stackrel{{\cal L}}{\longrightarrow}
G_{1,\alpha}$](sfehtmlimg3522.gif)
for
There is a tight relationship between the asymptotic behavior of
the maxima of random variables and the distribution of the
corresponding excesses which builds the foundation for an
important estimation method in the extreme value statistic, which
is defined in the next section. In general it deals with
observations crossing a specified threshold
. Their
distribution
is defined as follows:
Definition 18.5 (Excess Distribution)
Let
![$ u$](sfehtmlimg500.gif)
be an arbitrary threshold and
![$ F$](sfehtmlimg58.gif)
a distribution function
of an unbounded random variable
![$ X$](sfehtmlimg45.gif)
.
- a)
-
is called the
excess distribution beyond the threshold
.
- b)
-
, is the
average excess function.
With partial integration it follows that this definition of the
average excess function together with the following Theorem
17.4 agrees with:
If
is a random variable with the distribution function
then its expectation is
Theorem 18.5
![$ X$](sfehtmlimg45.gif)
is a positive, unbounded random variable with an absolutely
continuous distribution function
![$ F$](sfehtmlimg58.gif)
.
a) The average excess function
![$ e(u)$](sfehtmlimg3531.gif)
identifies
![$ F$](sfehtmlimg58.gif)
exactly:
b) If
![$ F$](sfehtmlimg58.gif)
is contained in the
MDA of the Fréchet distribution
![$ G_{1,\alpha}$](sfehtmlimg3475.gif)
, then
![$ e(u)$](sfehtmlimg3531.gif)
is
approximately linear for
![$ u \to \infty$](sfehtmlimg3533.gif)
:
Definition 18.6 (Pareto Distribution)
The
generalized Pareto distribution (GP =
generalized Pareto) with parameters
![$ \beta > 0,\ \gamma $](sfehtmlimg3535.gif)
has the distribution function
![$\displaystyle W_{\gamma, \beta} (x) = 1 - (1+ \frac{\gamma x}{\beta} )^ {-\frac{1}{\gamma}} \,
\; \; \;$](sfehtmlimg3536.gif)
for
and
![$ W_\gamma (x) = W_{\gamma, 1} (x) $](sfehtmlimg3539.gif)
is called the
generalized
standard Pareto distribution or
standardized GP
distribution.
Figure 17.3 shows the generalized standard Pareto distribution
with parameters
and
respectively.
For
the standardized GP distribution is an exponential distribution with parameter
For
it is a Pareto distribution
with
the parameter
. For
the GP
distribution is also referred to as a Beta distribution and
has the distribution function
.
Fig.:
Standard pareto distribution (
) with parameter
(red),
0 (black) and
(blue).
SFEgpdist.xpl
|
Theorem 18.6
The distribution
![$ F$](sfehtmlimg58.gif)
is contained in the MDA of the GEV
distribution
![$ G_\gamma$](sfehtmlimg3551.gif)
with the form parameter
![$ \gamma \ge 0$](sfehtmlimg3552.gif)
,
exactly when for a measurable function
![$ \beta (u) > 0$](sfehtmlimg3553.gif)
and the GP
distribution
![$ W_{\gamma, \beta}$](sfehtmlimg3554.gif)
it holds that:
![$\displaystyle \sup _{x \ge 0} \vert F_u (x) - W_{\gamma, \beta (u)} (x) \vert
\to 0 \,$](sfehtmlimg3555.gif)
for
A corresponding result also holds for the case when
,
in which case the supremum of
must be taken for those
.
For the generalized Pareto distribution
it holds for every finite threshold
![$\displaystyle F_u (x) = W_{\gamma, \beta + \gamma u} (x)$](sfehtmlimg3560.gif)
for
In this case
.