Extreme value theory of R language: visualization of tail index parameter estimation of hill statistics

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There are two main methods to estimate the extreme value index of sample tail distribution by extreme value theory ： Semi parametric method and full Parameter method , The former is mainly based on the distribution tail Hill An estimate , The latter is mainly based on generalized Pareto distribution .

Hill of tail index HILL Statistical estimation . More specifically , If we see  , and  , Then Hill HILL It is estimated that   

 . then    Meet a certain consistency in a sense   , If  , namely   （ Under the additional assumption of convergence rate , ）. Besides , Under additional technical conditions

To illustrate this point , Consider the following code . First , Let's consider a Pareto survival function , And the related quantile function

> Q=fuction(p){unro(funion(x) S(x)-(1-p),loer=1,per=1e+9)$root}

We will consider a more complex survival function . This is the survival function and quantile function ,

> plot(u,Veie(Q)(u),type="l")

ad locum , We need to The quantile function generates a random sample from this distribution ,

> X=Vectorize(Q)(runif(n))

hill The statistics are here

> abline(h=alpha)

​

We can now generate thousands of random samples , And look at these estimators （ For some specific   Of ）.

> for(s in 1:ns){
+ X=Vectorize
+ H=hill
+ hilk=function(k) 
+ HilK[s,]=Vectorize
+ }

If we calculate the average ,

> plot(15*(1:10),apply(2,mean)

​​​​​​​

We get a series of estimators that can be considered unbiased .

Now? , Think about it , be in Fréchet Distribution does not mean  , and  , But it means

For some slowly varying functions  , Not necessarily constant ！ To understand what might happen , We must be a little more specific . This can only be done by looking at the nature of the survival function . hypothesis , Here are some auxiliary functions   

This （ just ） constant    In some way, it is related to the convergence rate of the ratio of survival function to power function .

More specifically , hypothesis

then , The second-order regular variation property is obtained by using  , then , If    Tends to infinity, too fast , Then there will be a deviation in the estimation .  If  , that , For some  ,

The intuitive explanation for this result is , If    Too big , And if the base distribution is not Completely   It's a Pareto distribution , So Hill's estimate is biased . That's what we mean

• If    Too big ,   It's a biased estimator
• If    Too small ,   Is an unstable estimator

（ The latter comes from the attribute of the sample mean ： More observation , The smaller the volatility of the mean ）.

Let's run some simulations to better understand what's happening . Use the previous code , Generating random samples with survival function is actually extremely simple

> Q=function(p){uniroot(function(x) S(x)-(1-p)}

If we use the code above .

hill hill become

​

> abline(h=alpha)

​

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But it is based on only one sample . Consider again thousands of samples , Let's see Hill How... Statistics ,

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So these estimates are （ Experience ） The average is

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