Delta method and quantile estimation

An alternative (to profile likelihood techniques) to derive confidence intervals is to use the delta method. Consider an estimator such that

http://freakonometrics.blog.free.fr/public/perso5/delta02.gif

then for any differentiable transformation

http://freakonometrics.blog.free.fr/public/perso5/delta03.gif

The proof of that result is based on Taylor’s expansion (see here or there for more details on the theory, or even on this blog – here, in French or there in English – for some codes in R). This can be used to derive an asymptotic confidence interval for a quantile. Consider the following dataset

> base1=read.table(
+ "http://freakonometrics.free.fr/danish-univariate.txt",
+ header=TRUE)
> library(evir)
> X=base1$Loss.in.DKM

It is possible to fit a Generalized Pareto distribution on observations above a given threshold,

http://freakonometrics.blog.free.fr/public/perso5/mef08.gif

In that case, if http://freakonometrics.blog.free.fr/public/perso5/GPD10.gif exceed the threshold out of a sample of size http://freakonometrics.blog.free.fr/public/perso5/GPD11.gif, the estimator of the quantile

http://freakonometrics.blog.free.fr/public/perso5/GPD2.gif

i.e. http://freakonometrics.blog.free.fr/public/perso5/GPD05.gif. Then

http://freakonometrics.blog.free.fr/public/perso5/GPD06.gif

whilehttp://freakonometrics.blog.free.fr/public/perso5/GPD07.gif

i.e. it is now possible to implement the delta-method to derive the asymptotic variance of the quantile, and also (asymptotic) confidence intervals.

> u=5
> GPD=gpd(X,u)
> theta=GPD$par.ests
> sigma=GPD$varcov
> k=GPD$n.exceed
> n=length(X)
> p=.975
> Q=u+theta[2]/theta[1]*((n*(1-p)/k)^(-theta[1])-1)
> nabla=c(-theta[2]/theta[1]^2*((1-p)^(-theta[1])-1)-
+ theta[2]/theta[1]*(1-p)^(-theta[1]*log(1-p)),
+ 1/theta[1]*((1-p)^(-theta[1])-1))
> variance=t(nabla)%*%sigma%*%nabla

Based on the assumption of normality, it is possible to derive confidence intervals, and to compare them with the one obtained in R,

> c(Q-1.96/sqrt(k)*sqrt(variance),
+   Q+1.96/sqrt(k)*sqrt(variance))
[1] 13.11562 16.82852
+ tailplot(gpd(X,5))
+ gpd.q(tailplot(gpd(X,5)), .975, ci.type =
+ "likelihood", ci.p = 0.95,like.num = 50)
Lower CI Estimate Upper CI
13.33329 14.97207 17.18094

One thought on “Delta method and quantile estimation”

  1. Hello. Thank you for your constructive website. I wanted to know more about the beta technique in the area of uncertainty and prediction. Can you please introduce me some suitable tutorials or programs which I can learn beta technique easily?
    Many Thanks for your kind help,
    Bests

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