In the MAT8595 course, we’ve seen yesterday Hill estimator of the tail index. To be more specific, we did see see that if , with
, then Hill estimators for
are given by
for . Then we did say that
satisfies some consistency in the sense that
if
, but not too fast, i.e.
(under additional assumptions on the rate of convergence, it is possible to prove that
). Further, under additional technical conditions
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))
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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 ‘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 , with
, but is means that
for some slowly varying function , 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
such that
This (positive) constant 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
then, the second order regular variation property is obtained using , and then, if
goes to infinity too fast, then the estimator will be biased. More precisely (see Chapter 6 in Embrechts et al. (1997)), if
, then, for some
,
The intuitive interpretation of this result is that if 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
is too large,
is a biased estimator
- if
is too small,
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
> 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



, a Generalized Pareto Distribution will fit nicely, then we can use it to derive an estimator of the quantile function (for percentages such that the quantile is larger than the threshold)
, i.e. we keep the
largest observations to fit a GPD, then this estimator can be written








, where
is a slowly varying function. Then, for all
,







, so that
, where
is some slowly varying function. Equivalently, the exists a slowly varying function
such that
. Then
is the order statistic
, the slope of the straight line is the opposite of tail index
. The estimator of the slope is (considering only the
largest observations)
, as
), i.e.
, but not two fast, i.e.
(one can even get
with stronger convergence assumptions). Further

, then
. And similarly
as
with additional assumptions on the rate of convergence), and










of an i.i.d. sample
had a finite mean (based on extreme value results). Since I just used it on a small dataset (yes, with real data), I decided to post the R code, since it is rather simple to use. But instead of working on that dataset, let us see what happens on simulated samples. Consider
=200 observations generated from a Pareto distribution
=2, as a start)
>1.
(against the assumption that the expected value is finite, i.e.
), it is natural to consider the likelihood ratio
should be chi square distribution with one degree of freedom. As mentioned
, and thus, to fit a GPD on the
largest values. And then to plot all that on a graph (like the Hill plot)
the likelihood ratio statistics has a chi-square distribution).










Tous les blogs économiques saluent la parution en poche du joli petit livre de Daniel Zajdenweber, Economie des Extrêmes. En particulier, beaucoup de monde salue ce livre qui explique simplement des choses complexes…. Par exemple Alexandre dès 2001 “

















