18.1 Limit Behavior of Maxima

Consider the stochastic behavior of the maximum $M_n = \max (X_1, \ldots, X_n)$ of $n$ identically distributed random variables $X_1,\ldots,X_n$ with cumulative distribution function (cdf) $F(x)$. From a risk management perspective $X_t = -Z_t$ is the negative return at day $t$. The cdf of $M_n$ is

$${\P}(M_n \le x) = {\P}(X_1 \le x, \ldots, X_n \le x) = \prod_{t=1}^n {\P}(X_t \le x) = F^ n (x).$$ (18.1)

We are only considering unbounded random variables $X_t$, i.e. $F(x) < 1$ for all $x < \infty.$ Obviously it holds that $F^n(x) \to 0$ for all $x$, when $n\to \infty$, and thus $M_n \stackrel{P}{\longrightarrow} \infty$. The maximum of $n$ unbounded random variables increases over all boundaries. In order to achieve a non-degenerate behavior limit, $M_n$ has to be standardized in a suitable fashion.

Definition 18.1 (Maximum Domain of Attraction)  
The random variable $X_t$ belongs to the maximum domain of attraction (MDA) of a non-degenerate distribution $G$, if for suitable sequences $c_n > 0, d_n$ it holds that:

$$\frac{M_n - d_n}{c_n} \stackrel{{\cal L}}{\longrightarrow} G$$   for $$\, \, n \to \infty,$$

i.e. $F^ n (c_n x + d_n) \to G(x)$ at all continuity points $x$ of the cdf $G(x)$.

It turns out that only a few distributions $G$ can be considered as the asymptotic limit distribution of the standardized maximum $M_n$. They are referred to as the extreme value distriubtions. These are the following three distribution functions:

Fréchet:	$G_{1,\alpha} (x) = \exp \{ - x^{- \alpha} \},\ x \ge 0\ ,$ for $\alpha > 0 ,$
Gumbel:	$G_0 (x) = \exp \{ - e^ {-x}\},\ x \in \mathbb{R} ,$
Weibull:	$G_{2,\alpha} (x) = \exp\{ - \vert x\vert^ {-\alpha} \},\ x \le 0\ ,$ for $\alpha < 0 .$

Fig.: Fréchet (red), Gumbel (black) and Weibull distributions (blue). SFEevt1.xpl

The Fréchet distributions are concentrated on the non-negative real numbers $[0,\infty)$, while the Weibull distribution, on the other hand, on $(- \infty, 0],$ 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 $\alpha=2$ and the Weibull distribution with parameter $\alpha=-2$. 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 $\gamma \in \mathbb{R}$ has the distribution function:

	$G_\gamma (x) = \exp \{ - (1+\gamma x)^ {- 1/\gamma} \},\ 1 + \gamma x > 0\,$ for $\gamma \not= 0$
	$G_0 (x) = \exp \{ - e^ {-x}\},\ x \in \mathbb{R}$

$G_0$ is the Gumbel distribution, whereas $G_\gamma, \gamma \neq 0$ is linked to the Fréchet- and Weibull distributions by the following relationships:

$$G_\gamma (\frac{x-1}{\gamma}) = G _{1,1/\gamma} (x)\$$   for $$\gamma > 0,$$

$$G_\gamma (- \frac{x+1}{\gamma} ) = G_{2,- 1/\gamma} (x)$$   for $$\gamma < 0.$$

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: $G(x) = G_\gamma (\frac{x-\mu}{\sigma})$ with the form parameter $\gamma$, the location parameter $\mu \in \mathbb{R}$ and the scale parameter $\sigma > 0.$ For asymptotic theory this does not matter since the standardized sequences $c_n, d_n$ can be always chosen so that the asymptotic distribution $G$ has the standard form ( $\mu = 0, \sigma = 1$). An important result of the asymptotic distribution of the maximum $M_n$ is the Fisher-Tippett theorem:

Theorem 18.1  
If there exists sequences $c_n > 0, d_n$ and a non-degenerate distribution $G$, so that

$$\frac{M_n - d_n}{c_n} \stackrel{{\cal L}}{\longrightarrow} G$$   for $$n \to \infty,$$

then $G$ is a GEV distribution.

Proof:
As a form of clarification the basic ideas used to prove this central result are outlined. Let $t>0$, and $[ z ]$ represent the integer part of $z$. Since $F^ {[nt]}$ is the distribution function of $M_{[nt]}$, due to our assumptions on the asymptotic distribution of $M_n$ it holds that

$$F^ {[nt]} ( c_{[nt]} x + d_{[nt]} ) \longrightarrow G(x) \,$$    for $$\, [nt] \rightarrow \infty,\$$    i.e.  $$n\rightarrow \infty.$$

On the other hand it also holds that

$$F^ {[nt]} (c_n x + d_n) = \{ F^ n (c_n x + d_n) \} ^ {\frac{[nt]}{n}} \longrightarrow G^ t (x)\$$   for $$n\rightarrow \infty.$$

In other words this means that

$$\frac{M_{[nt]} - d_{[nt]}}{c_{[nt]}} \stackrel{{\cal L}}{\longrig... ...\ \, \quad \frac{M_{[nt]} - d_n}{c_n} \stackrel{{\cal L}}{\longrightarrow} G^ t$$

for $n\to \infty$. According to the Lemma, which is stated below, this is only possible when

$$\frac{c_n}{c_{[nt]}} \longrightarrow b(t) \ge 0,\ \quad \frac{d_n - d_{[nt]}}{c_{[nt]}} \longrightarrow a(t)$$

and

$$G^ t(x) = G(b(t) x + a(t) ) ,\ t > 0,\ x \in \mathbb{R}.$$ (18.2)

This relationship holds for arbitrary values $t$. We use it in particular for arbitrary $t, s$ and $s\cdot t$ and obtain

$$b (st) =b (s)\ b (t),\ a (st) = b (t) a (s)+ a (t).$$ (18.3)

The functional equations (17.2), (17.3) for $G(x), b (t), a (t)$ have only one solution, when $G$ is one of the distributions $G_0, G_{1,\alpha}$ or $G_{2,\alpha}$, that is, $G$ must be a GEV distribution.
${\Box}$

Lemma 18.1 (Convergence Type Theorem)  
Let $U_1, U_2, \ldots, V, W$ be random variables, $b_n, \beta _n > 0,\ a_n, \alpha_n \in \mathbb{R}.$ If

$$\frac{U_n - a_n}{b_n} \stackrel{{\cal L}}{\longrightarrow} V$$

in distribution for $n\to \infty$, then it holds that:

$$\frac{U_n - \alpha_n}{\beta_n} \stackrel{{\cal L}}{\longrightarrow} W\$$   if and only if $$\hspace{0.5 cm} \frac{b_n}{\beta_n} \longrightarrow b \ge 0,\ \frac{a_n - \alpha_n}{\beta_n} \longrightarrow a \in \mathbb{R}.$$

In this case $W$ has the same distribution as $bV+a$.

Notice that the GEV distributions are identical to the so called max-stable distributions, by which for all $n \ge 1$ the maximum $M_n$ of $n$ i.i.d. random variables $X_1,\ldots,X_n$ have the same distribution as $c_n X_1 + d_n$ for appropriately chosen $c_n > 0, d_n$.

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 $F(x)=\Phi(x)$ with computer generated random variables that have a Gumbel distribution, Fréchet distribution with parameter $\alpha=2$ and a Weibull distribution with parameter $\alpha=-2$ 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 $x$, when $x$ increases. Since this exceedance probability plays an important role in extreme value theory, we will introduce some more notations:

$$\overline{F}(x) = {\P}(X_1 > x) = 1 - F(x).$$

The relationship between the exceedance probability $\overline{F}(x)$ and the distribution of the maxima $M_n$ will become clear with the following theorem.

Theorem 18.2  
a) For $0 \le \tau \le \infty$ and every sequence of real numbers $u_n, n \ge 1,$ it holds for $n\to \infty$ that

$$n\overline{F} (u_n) \to \tau \, \; \;$$   if and only if  $$\; \; {\P}(M_n \le u_n) \to e^ {- \tau}.$$

b) $F$ belongs to the maximum domain of attraction of the GEV distribution $G$ with the standardized sequences $c_n, d_n$ exactly when $n\to \infty$

$$n \overline{F} (c_n x + d_n) \to - \log G(x)$$   for all $$\, x \in \mathbb{R}.$$

The exceedance probability of the Fréchet distribution $G_{1,\alpha}$ behaves like $1/x^\alpha$ for $x \to \infty$, because the exponential function around 0 is approximately linear, i.e.,

$$\overline{G}_{1,\alpha}(x) = \frac{1}{x^ \alpha} \{1 + {\scriptstyle \mathcal{O}}(1) \}$$   for$$\quad x \to \infty.$$

Essentially all of the distributions that belong to the MDA of this Fréchet distribution show the same behavior; $x^\alpha \overline{F}(x)$ is almost constant for $x \to \infty$, or more specifically: a slowly varying function.

Definition 18.3  
A positive measurable function $L$ in $(0,\infty)$ is called slowly varying, if for all $t>0$

$$\frac{L(tx)}{L(x)} \to 1$$   for$$\quad x \to \infty.$$

Typical slowly varying functions are, in addition to constants, logarithmic growth rates, for example $L(x)=\log(1+x), x>0$.

Theorem 18.3  
$F$ belongs to the maximum domain of attraction of the Fréchet distribution $G_{1,\alpha}$ for some $\alpha>0$, if and only if $x^ \alpha \overline{F}(x) = L(x)$ is a slowly varying function. The random variables $X_t$ with the distribution function $F$ are unbounded, i.e., $F(x) < 1$ for all $x < \infty,$ and it holds that

$$\frac{M_n}{c_n} \stackrel{{\cal L}}{\longrightarrow} G_{1,\alpha}$$

with $c_n = F^{-1}(1 - \frac{1}{n}).$

For the description of the standardized sequence $c_n$ we have used the following notation. $c_n$ is an extreme quantile of the distribution $F$, and it holds that $\overline{F}(c_n) = {\P}(X_t > c_n) = 1/n.$

Definition 18.4 (Quantile Function)  
If $F$ is a distribution function, we call the generalized inverse

$$F^{-1}(\gamma) = \inf \{ x \in \mathbb{R};\ F(x) \ge \gamma \},\ \, 0 < \gamma < 1,$$

the quantile function. It then holds that ${\P}( X_1 \le F^{-1}(\gamma ) )=\gamma$, i.e., $F^{-1}(\gamma )$ is the $\gamma$-quantile of the distribution $F$.

If $F$ is strictly monotonic increasing and continuous, then $F^{-1}$ is the generalized inverse of $F$.

There is a corresponding criterion for the Weibull distribution that can be shown using the relationship $G_{2,\alpha} (- x^{-1}) = G_{1,\alpha} (x),\ x > 0,$. Random variables, whose maxima are asymptotically Weibull distributed, are by all means bounded, i.e., there exists a constant $c < \infty$, such that $X_t \le c$ 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 $x$ only slowly decreases with $x$, 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 $F$ of the unbounded random variables $X_t$ belongs to the maximum domain of attraction of the Gumbel distribution if measurable scaling functions $c(x), g(x) > 0$ as well as an absolute continuous function $e(x) > 0$ exist with $c(x) \to c > 0, \ g(x) \to 1, e'(x) \to 0$ for $x \to \infty$ so that for $z < \infty$

$$\overline{F}(x) = c(x) \exp \{ - \int^ x_z \, \frac{g(y)}{e(y)} dy \},\ z < x < \infty.$$

In this case it holds that

$$\frac{M_n-d_n}{c_n} \stackrel{{\cal L}}{\longrightarrow} G_0$$

with $d_n = F^{-1}(1- \frac{1}{n})$ and $c_n = e(d_n).$

As a function $e(x)$, the average excess function can be used:

$$e(x) = \frac{1}{\overline{F}(x)} \int^ {\infty}_x \overline{F} (y) \, dy,\ \, x < \infty,$$

which will be considered in more detail in the following.

The exponential distribution with parameter $\lambda$ has the distribution function
$F(x)=1-e^{-\lambda x}, x \ge 0,$ so that $\overline{F} (x) = e^{-\lambda x}$ fulfills the conditions stipulated in the theorem with $c(x)= 1,\ g(x) = 1,\ z = 0$ and $e(x) = 1 / \lambda$. The maximum $M_n$ of $n$ independent exponentially distributed random variables with parameter $\lambda$ thus converges in distribution to the Gumbel distribution:

$$\lambda (M_n - \frac{1}{\lambda} \ \log \ n) \stackrel{{\cal L}}{\longrightarrow} G_0$$   for$$\quad n \to \infty.$$

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, $M_n$ is the maximum of $n$ independent standard normally distributed random variables, then it holds that

$$\sqrt{2\ \log \ n} (M_n - d_n) \stackrel{{\cal L}}{\longrightarrow} G_0 \quad n\rightarrow\infty$$

$$\quad d_n = \sqrt{2 \ \log \ n} - \frac{ \log \log \ n + \log (4\pi)}{ 2 \sqrt{2 \ \log \ n}}.$$

SFEevtex1.xpl

Another member of the distributions in the maximum domain of attraction of the Fréchet distribution $G_{1,\alpha}$ is the Pareto distribution with the distribution function

$$W_{1,\alpha}(x)=1-\frac{1}{x^\alpha}, x \ge 1, \alpha > 0,$$

as well as all other distributions with Pareto tails, i.e., with

$$\overline{F}(x) = \frac{\kappa}{x^ \alpha} \{ 1 + {\scriptstyle \mathcal{O}}(1) \} \; \; \;$$   for$$\; \; \; x \to \infty.$$

Since $\overline{F}^{-1}(\gamma)$ for $\gamma \approx 1$ behaves here like $(\kappa/\gamma)^{1/\alpha}$, $c_n$ for $n\to \infty$ is identical to $(\kappa n)^ {1/\alpha}$, and

$$\frac{M_n}{(\kappa n)^ {1/\alpha}} \stackrel{{\cal L}}{\longrightarrow} G_{1,\alpha}$$   for $$\, n \to \infty.$$

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 $u$. Their distribution $F_u$ is defined as follows:

Definition 18.5 (Excess Distribution)  
Let $u$ be an arbitrary threshold and $F$ a distribution function of an unbounded random variable $X$.

a)

$F_u (x) = {\P}\{ X - u \le x \ \vert \ X> u\} = \{F(u+x) - F(u)\}/\overline{F} (u),\ 0 \le x < \infty$ is called the excess distribution beyond the threshold $u$.

b)

$e(u) = {\mathop{\text{\rm\sf E}}} \{ X-u \ \vert \ X> u\},\ \, 0 < u < \infty$, 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:

$$e(u) = \int^ {\infty}_u \frac{\overline{F}(y)}{\overline{F}(u)} dy.$$

If $\Delta_u$ is a random variable with the distribution function $F_u,$ then its expectation is ${\mathop{\text{\rm\sf E}}} \Delta_u = e(u).$

Theorem 18.5  
$X$ is a positive, unbounded random variable with an absolutely continuous distribution function $F$.
a) The average excess function $e(u)$ identifies $F$ exactly:

$$\overline{F} (x) = \frac{e(0)}{e(x)} \exp \{ - \int^ x_0 \frac{1}{e(u)} du \},\ \, x > 0.$$

b) If $F$ is contained in the MDA of the Fréchet distribution $G_{1,\alpha}$, then $e(u)$ is approximately linear for $u \to \infty$: $e(u) = \frac{1}{\alpha -1} \ u \{ 1 + {\scriptstyle \mathcal{O}}(1) \}.$

Definition 18.6 (Pareto Distribution)  
The generalized Pareto distribution (GP = generalized Pareto) with parameters $\beta > 0,\ \gamma$ has the distribution function

$$W_{\gamma, \beta} (x) = 1 - (1+ \frac{\gamma x}{\beta} )^ {-\frac{1}{\gamma}} \, \; \; \;$$   for$$\, \, \; \; \left\{ \begin{array}{lll} x \ge 0 & \text{\rm if} & ... ...x \le \frac{- \beta}{\gamma} & \text{\rm if} & \gamma < 0, \end{array} \right.$$

and

$$W_{0,\beta } (x) = 1 - e^ {-\frac{1}{\beta} x},\ x \ge 0.$$

$W_\gamma (x) = W_{\gamma, 1} (x)$ is called the generalized standard Pareto distribution or standardized GP distribution.

Figure 17.3 shows the generalized standard Pareto distribution with parameters $\gamma = 0.5, 0$ and $-0.5$ respectively.

For $\gamma = 0$ the standardized GP distribution is an exponential distribution with parameter $1.$ For $\gamma>0$ it is a Pareto distribution $W_{1,\alpha}$ with the parameter $\alpha = 1/\gamma$. For $\gamma < 0$ the GP distribution is also referred to as a Beta distribution and has the distribution function $W_{2,\alpha}=1-(-x)^{-\alpha}, -1 \le x \le 0, \alpha<0$.

Fig.: Standard pareto distribution ($\beta=1$) with parameter $\gamma = 0.5$ (red), 0 (black) and $-0.5$ (blue). SFEgpdist.xpl

Theorem 18.6  
The distribution $F$ is contained in the MDA of the GEV distribution $G_\gamma$ with the form parameter $\gamma \ge 0$, exactly when for a measurable function $\beta (u) > 0$ and the GP distribution $W_{\gamma, \beta}$ it holds that:

$$\sup _{x \ge 0} \vert F_u (x) - W_{\gamma, \beta (u)} (x) \vert \to 0 \,$$   for $$\, u \to \infty.$$

A corresponding result also holds for the case when $\gamma < 0$, in which case the supremum of $x$ must be taken for those $0 < W_{\gamma, \beta (u)} (x) < 1$.

For the generalized Pareto distribution $F = W_{\gamma, \beta}$ it holds for every finite threshold $u > 0$

$$F_u (x) = W_{\gamma, \beta + \gamma u} (x)$$   for$$\quad \left\{ \begin{array}{lll} x \ge 0 & \text{\rm if} & \gamma... ...< - \frac{\beta}{\gamma} - u & \text{\rm if} & \gamma < 0, \end{array} \right.$$

In this case $\beta (u) = \beta + \gamma \, u$.