Detecting distributions with infinite mean

In a post I published a few month ago (in French, here, based on some old paper,there), I mentioned a statistical procedure to test if the underlying distribution https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO02.pngof an i.i.d. sample https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO01.png 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 https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO03.png=200 observations generated from a Pareto distribution

https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO04.png

(here https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png=2, as a start)

> a=2
> X=runif(200)^(-1/a)

Here, we will use the package developped by Mathieu Ribatet,

> library(RFA)
  • Using Generalized Pareto Distribution (and LR test)

A first idea is to fit a GPD distribution on observations that exceed a threshold https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO11.png>1.
Since we would like to test https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO06.png (against the assumption that the expected value is finite, i.e. https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO07.png), it is natural to consider the likelihood ratio

https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO08.png

Under the null hypothesis, the distribution of https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO09.png should be chi square distribution with one degree of freedom. As mentioned here, the significance level is attained with a higher accuracy by employing Bartlett correction (there). But let  us make it as simple as possible for the blog, and use the chi-square distribution to derive the p-value.
Since it is rather difficult to select an appropriate threshold, it can be natural (as in Hill’s estimator) to consider https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO10.png, and thus, to fit a GPD on the https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO13.pnglargest values. And then to plot all that on a graph (like the Hill plot)

> Xs=rev(sort(X))
> s=0;G=rep(NA,length(Xs)-14);Gsd=G;LR=G;pLR=G
> for(i in length(X):15){
+ s=s+1
+ FG=fitgpd(X,Xs[i],method="mle")
+ FGc=fitgpd(X,Xs[i],method="mle",shape=1)
+ G[s]=FG$estimate[2]
+ Gsd[s]=FG$std.err[2]
+ FGc$fixed
+ LR[s]=FGc$deviance-FG$deviance
+ pLR[s]=1-pchisq(LR[s],df=1)
+ }

Here we keep the estimated value of the tail index, and the associated standard deviation of the estimator, to draw some confidence interval (assuming that the maximum likelihood estimate is Gaussian, which is correct only when n is extremely large). We also calculate the deviance of the model, the deviance of the constrained model (https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO06.png), and the difference, which is the log likelihood ratio. Then we calculate the p-value (since under https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO12.png the likelihood ratio statistics has a chi-square distribution).
If https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png=2, we have the following graph, with on top the p-value (which is almost null here), the estimation of the tail index he https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO13.png largest values (and a confidence interval for the estimator),

If https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png=1.5 (finite mean, but infinite variance), we have

i.e. for those two models, we clearly reject the assumption of infinite mean (even if https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png gets too close from 1, we should consider thresholds large enough). On the other hand, if https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png=1 (i.e. infinite mean) we clearly accept the assumption of infinite mean (whatever the threshold),

  • Using Hill’s estimator

An alternative could be to use Hill’s estimator (with Alexander McNeil’s package). See here for more details on that estimator. The test is simply based on the confidence interval derived from the (asymptotic) normal distribution of Hill’s estimator,

> library(evir)
> Xs=rev(sort(X))
> HILL=hill(X)
> G=rev(HILL$y)
> Gsd=rev(G/sqrt(HILL$x))
> pLR=1-pnorm(rep(1,length(G)),mean=G,sd=Gsd)

Again, if https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png=2, we clearly rejct the assumption of infinite mean,

and similarly, if https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png=1.5 (finite mean, but infinite variance)

Here the test is more robust than the one based on the GPD. And if https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png=1 (i.e. infinite mean), again we accept https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO12.png,

Note that if https://perso.univ-rennes1.fr/arthur.charpentier/latex/paretoRO05.png=0.7, it is still possible to run the procedure, and hopefully, it suggests that the underlying distribution has infinite mean,

(here without any doubt). Now you need to wait a few days to see some practical applications of the idea (there was on in the paper mentioned above actually, on business interruption insurance losses).


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