# Freakonometrics

## Tag - ratio

Tuesday, February 21 2012

## Tests on tail index for extremes

Since several students got the intuition that natural catastrophes might be non-insurable (underlying distributions with infinite mean), I will post some comments on testing procedures for extreme value models.

A natural idea is to use a likelihood ratio test (for composite hypotheses). Let denote the parameter (of our parametric model, e.g. the tail index), and we would like to know whether is smaller or larger than (where in the context of finite versus infinite mean ). I.e. either belongs to the set or to its complementary . Consider the maximum likelihood estimator , i.e.

Let and denote the constrained maximum likelihood estimators on and respectively,

Either and (on the left), or and (on the right)

So likelihood ratios

are either equal to

or

If we use the code mentioned in the post on profile likelihood, it is easy to derive that ratio. The following graph is the evolution of that ratio, based on a GPD assumption, for different thresholds,
```> base1=read.table(
+ "http://freakonometrics.free.fr/danish-univariate.txt",
> library(evir)
> X=base1\$Loss.in.DKM
> U=seq(2,10,by=.2)
> LR=P=ES=SES=rep(NA,length(U))
> for(j in 1:length(U)){
+ u=U[j]
+ Y=X[X>u]-u
+ loglikelihood=function(xi,beta){
+ sum(log(dgpd(Y,xi,mu=0,beta))) }
+ XIV=(1:300)/100;L=rep(NA,300)
+ for(i in 1:300){
+ XI=XIV[i]
+ profilelikelihood=function(beta){
+ -loglikelihood(XI,beta) }
+ L[i]=-optim(par=1,fn=profilelikelihood)\$value }
+ plot(XIV,L,type="l")
+ PL=function(XI){
+ profilelikelihood=function(beta){
+ -loglikelihood(XI,beta) }
+ return(optim(par=1,fn=profilelikelihood)\$value)}
+ (L0=(OPT=optimize(f=PL,interval=c(0,10)))\$objective)
+ profilelikelihood=function(beta){
+ -loglikelihood(1,beta) }
+ (L1=optim(par=1,fn=profilelikelihood)\$value)
+ LR[j]=L1-L0
+ P[j]=1-pchisq(L1-L0,df=1)
+ G=gpd(X,u)
+ ES[j]=G\$par.ests[1]
+ SES[j]=G\$par.ses[1]
+ }
>
> plot(U,LR,type="b",ylim=range(c(0,LR)))
> abline(h=qchisq(.95,1),lty=2)```

with on top the values of the ratio (the dotted line is the quantile of a chi-square distribution with one degree of freedom) and below the associated p-value
```> plot(U,P,type="b",ylim=range(c(0,P)))
> abline(h=.05,lty=2)```

In order to compare, it is also possible to look at confidence interval for the tail index of the GPD fit,
```> plot(U,ES,type="b",ylim=c(0,1))
> lines(U,ES+1.96*SES,type="h",col="red")
> abline(h=1,lty=2)```

To go further, see Falk (1995), Dietrich, de Haan & Hüsler (2002), Hüsler & Li (2006) with the following table, or Neves & Fraga Alves (2008). See also here or there (for the latex based version) for an old paper I wrote on that topic.

Tuesday, May 18 2010

## Ratio moyen et moyenne de ratios

Pour continuer dans le même esprit que le billet sur les moyennes, ici, un petit billet sur les ratios (le ratio est un concept fondamental dans la règlementation bancaire et financière). Dans une vie antérieure, un manager d'une compagnie d'assurance m'avait dit que le ratio moyen (en l'occurrence du montant des sinistres sur le total des primes encaissées) était la moyenne des ratio. Si ce résultat est faux, i.e.

il n'en reste pas moins que le ratio moyen est compris entre les valeurs extrêmes

si

La preuve en image,

ce qui montre que

quod erat demonstrandum.