# Bias of Hill Estimators

In the MAT8595 course, we’ve seen yesterday Hill estimator of the tail index. To be more specific, we did see see that if $\overline{F}(x)=C x^{-\alpha}$, with $\alpha>0$, then Hill estimators for $\alpha$ are given by

$\widehat{\alpha}_k = \left[\frac{1}{k}\sum_{i=0}^{k-1} \log X_{n-i,n} -\log X_{n-k,n}\right]^{-1}$
for $k\in\{1,2,\cdots,n\}$. Then we did say that $\widehat{\alpha}_k$ satisfies some consistency in the sense that $\widehat{\alpha}_k \overset{\mathbb{P}}{\rightarrow} \alpha$ if $k\rightarrow\infty$, but not too fast, i.e. $k/n\rightarrow0$ (under additional assumptions on the rate of convergence, it is possible to prove that $\widehat{\alpha}_k \overset{a.s.}{\rightarrow} \alpha$). Further, under additional technical conditions

$\sqrt{k}\left(\widehat{\alpha}_k-\alpha\right)\overset{\mathcal L}{\rightarrow}\mathcal{N}(0,\alpha^2)$

In order to illustrate this point, consider the following code. First, let us consider a Pareto survival function, and the associated quantile function

> alpha=1.5
> S=function(x){ifelse(x>1,x^(-alpha),1)}
> Q=function(p){uniroot(function(x) S(x)-(1-p),lower=1,upper=1e+9)$root} The code here is obviously too complicated, since this power function can easily be inverted. But later on, we will consider a more complex survival function. Here are the survival function, and the quantile function, > u=seq(0,5,by=.01) > plot(u,Vectorize(S)(u),type="l",col="red") > u=seq(0,99/100,by=.01) > plot(u,Vectorize(Q)(u),type="l",col="blue",ylim=c(0,20)) Here, we need the quantile function to generate a random sample from this distribution, > n=500 > set.seed(1) > X=Vectorize(Q)(runif(n)) Hill plot is here > library(evir) > hill(X) > abline(h=alpha,col="blue") We can now generate thousands of random samples, and see how those estimators behave (for some specific $k$‘s). > ns=10000 > HillK=matrix(NA,ns,10) > for(s in 1:ns){ + X=Vectorize(Q)(runif(n)) + H=hill(X,plot=FALSE) + hillk=function(k) H$y[H$x==k] + HillK[s,]=Vectorize(hillk)(15*(1:10)) + } and if we compute the average, > plot(15*(1:10),apply(HillK,2,mean) we do get a series of estimators that can be considered as unbiased. So far, so good. Now, recall that being in the max-domain of attraction of the Fréchet distribution does not mean that $\overline{F}(x)=C x^{-\alpha}$, with $\alpha>0$, but is means that $\overline{F}(x)= x^{-\alpha} \mathcal{L}(x)$ for some slowly varying function $\mathcal{L}$, not necessarily constant! In order to understand what could happen, we have to be slightly more specific. And this can be done only by looking at second order regular variation property of the survival function. Assume, here that there is some auxilary function $a$ such that $\lim_{t\rightarrow\infty}\frac{\overline{F}(xt)/\overline{F}(t)-x^{-\alpha}}{a(t)}=x^{-\alpha}\frac{1-x^{-\beta}}{\beta}{}$ This (positive) constant $\beta$ is – somehow – related to the speed of convergence of the ratio of the survival functions to the power function (see e.g. Geluk et al. (2000) for some examples). To be more specific, assume that $\overline{F}(x)=\underbrace{C(1+x^{-\beta})}_{\mathcal{L}(x)}\cdot x^{-\alpha}$ then, the second order regular variation property is obtained using $a(t)=\beta t^{-\beta}$, and then, if $k$ goes to infinity too fast, then the estimator will be biased. More precisely (see Chapter 6 in Embrechts et al. (1997)), if $k=O(n^{2\beta/(\alpha+2\beta)})$, then, for some $\lambda>0$, $\sqrt{k}\left(\widehat{\alpha}_k-\alpha\right)\overset{\mathcal L}{\rightarrow}\mathcal{N}\left(\frac{\alpha^3}{\beta-\alpha}\lambda,\alpha^2\right)$ The intuitive interpretation of this result is that if $k$ is too large, and if the underlying distribution is not exactly a Pareto distribution (and we do have this second order property), then Hill’s estimator is biased. This is what we mean when we say • if $k$ is too large, $\widehat{\alpha}_k$ is a biased estimator • if $k$ is too small, $\widehat{\alpha}_k$ is a volatile estimator (the later comes from properties of a sample mean: the more observations, the less the volatility of the mean). Let us run some simulations to get a better understanding of what’s going on. Using the previous code, it is actually extremly simple to generate a random sample with survival function $\overline{F}(x)=\underbrace{C(1+x^{-\beta})}_{\mathcal{L}(x)}\cdot x^{-\alpha}$ > beta=.5 > S=function(x){+ ifelse(x>1,.5*x^(-alpha)*(1+x^(-beta)),1) } > Q=function(p){uniroot(function(x) S(x)-(1-p),lower=1,upper=1e+9)$root}

If we use the code above. Here, with

> n=500
> set.seed(1)
> X=Vectorize(Q)(runif(n))

the Hill plot becomes

> library(evir)
> hill(X)
> abline(h=alpha,col="blue")

But it’s based on one sample, only. Again, consider thousands of samples, and let us see how Hill’s estimator is behaving,

so that the (empirical) mean of those estimator is

# MAT8886 Extremes and sums (of i.i.d. random variables)

Yesterday, we have discussed briefly sums and maximas of i.i.d. random variables using the concept of subexponential distributions. Today, we will introduce the concept of regular variation: a positive function is said to be regularly varying (at infinity), denoted , for some , if

for all . An this concept can be related to sums and maxima (see section 6.2.6 in Embrechts et al. (1997)). Consider i.i.d. positive random variables : let and . Then it can be shown easily that

•  if and only if

•  for some  if and only if the exists a non-degenerate variable  such that

•  with  if and only if

If is not that simple to check for such convergences, it is still possible to use graphs to study the behavior of the empirical version of those quantities. Consider the following function to visualize convergence of empirical ratios,

CONVERGENCE=function(g,p=1,n=500000){
set.seed(1)
X=g(n);X1=g(n);X2=g(n);X3= g(n);X4=g(n)
Tp =cummax(X^p)/cumsum(X^p)
Tp1=cummax(X1^p)/cumsum(X1^p)
Tp2=cummax(X2^p)/cumsum(X2^p)
Tp3=cummax(X3^p)/cumsum(X3^p)
Tp4=cummax(X4^p)/cumsum(X4^p)
plot(Tp4,type="l",ylim=c(0,1),log="x",
xlim=c(100,n),ylab="",col="light blue",xlab="")
lines(Tp1,col="light green")
lines(Tp2,col="yellow")
lines(Tp3,col="pink")
lines(Tp,lwd=2)
abline(h=0:1,col="red",lty=2)
}

or the following to study the “asymptotic” distribution of the ratio on simulated samples

LIMITDIST=function(g,p=1,n=500000,ns=1000){
set.seed(1)
T=rep(NA,ns)
for(i in 1:ns){
X=g(n)
T[i]=max(X^p)/sum(X^p)
}
hist(T,breaks=seq(0,1,by=.05),probability=TRUE,
col="light green",ylab="",xlab="",main="")
}

In the case of exponentially distributed variables, we have

CONVERGENCE(rexp)

For variables with a lognormal distribution,

CONVERGENCE(rlnorm)

And finally, consider the case of a Pareto distribution

rpareto=function(n){runif(n)^(-1/1.5)-1}
CONVERGENCE(rpareto)

Here, it looks like those three distributions have finite variance (and actually, they do). To go one step further, for , define  and . Then analogous results can be derived,

•  if and only if

•  for some  if and only if the exists a non-degenerate variable  such that

•  with  if and only if

Again, it is possible to use the function defined above,

CONVERGENCE(rexp,p=2)

or

CONVERGENCE(rexp,p=3)

or even

CONVERGENCE(rexp,p=10)

If the power is not too high, it looks like the ratio goes to zero. But when it becomes larger, it looks like more simulations might be necessary to say something relevant.

CONVERGENCE(rlnorm,p=2)

or

CONVERGENCE(rlnorm,p=3)

Here also, it looks like we have a light tailed distribution (and actually, it is the case). And finally, if we consider the case of a Pareto distribution

CONVERGENCE(rpareto,p=2)

Then it looks like it is an heavy tailed distribution. In order to get a better understanding, plot the distribution of the ratio obtained from 1,000 simulated samples (of size 500,000),

LIMITDIST(rpareto,p=1)

versus

LIMITDIST(rpareto,p=2)

So obviously, something is going on between 1 and 2 (recall that the power parameter of the Pareto distribution is 1.5).

# Fisher-Tippett theorem with an historical perspective

A couple of weeks ago, Rafael asked me if I had something on the history of extreme value theory. Since I will get back to fundamental results about extremes in my course, I promised I will write down a short post on all that issue.

To start from the beginning, in 1928, Ronald Fisher and Leonard Tippett formulated the three types of limiting distributions for the maximum term of a random sample (Fisher & Tippett (1928)). The problem was to characterize function such that

where where ‘s are i.i.d. with cumulative distribution function . They had supporting arguments, but no (rigorous) proof. Nevertheless, the obtained that the only possible types for G were

i.e. Fréchet type (Pareto-type tails), or

i.e. Weibull type (bounded distribution type), or

i.e. Gumbel type (exponential-type tails). Emil Gumbel has been intensively using the so-called Gumbel distribution on river flows, since (as he explained in 1958), “it seems that the rivers know the theory. It only remains to convince the engineers of the validity of this analysis“.
Independently of that work (published in 1928), Maurice Fréchet considered in 1927 (in Sur la loi de probabilité de l’écart maximum) possible limits of

and obtained only as possible limit. Richard von Mises gave in 1936 sufficient, but not necessary conditions for their (max) domain of attraction, i.e. characterization of function such that the maxima converges to some specific function (von Mises (1936)). E.g. he noticed that a sufficient condition on to be in the (max) domain of attraction of the Gumbel distribution is that

Then in 1943, Boris Gnedenko gave a complete characterization of those three types, with a complete characterization for two of them (heavy tails, i.e. Fréchet type and bounded support, i.e. Weibull) but his necessary and sufficient condition was based on a function that was not explicitly defined (see Gnedenko (1943)). Laurens de Haan in the 70’s derived checkable condition for Gumbel’s type.
Boris Gnedenko proved (in Section 4 of his paper) that F is the (max) domain of attraction of if and only if is regularly varying at infinity, with index (even if the term “regular variation” was not mentioned in the paper). Similar results were derived to characterize functions in the (max) domain of attraction of Weibull. For the (max) domain of attraction of , Boris Gnedenko obtained that a necessary and sufficient condition was that there exists a function such goes to 0 at infinity and

Several papers have discussed what function could be e.g. David Mejzler in 1949 (in Russian, but see also his 1965 paper), and Laurens de Hann in 1970 and 1971 (following the dramatic flood in the Netherlands in 1953, researchers in the Netherlands have focuses on dikes, and extreme value applications).

Mejzler’s idea was to work on quantiles, and not on the cumulative distribution function. I.e. define

Then a necessary and sufficient condition for F to be in the (max) domain of attraction of is that

Laurens de Haan proved in 1971 that function can be – in general – given by

And in 1976, Laurens de Haan obtained a three-type convergence working on quantile function (with a much shorter proof).
There have been many many papers extending Fisher-Tippett’s theorem, e.g. on non-independent sequences, like exchangeable ones (in a paper by Simeon Berman in 1962, or on stationary Gaussian sequences in 1964).