Detecting tail behavior: mean excess plots with confidence bounds

TL;DR

This paper develops confidence bounds for mean excess plots to help identify the tail behavior of data, distinguishing among Fréchet, Gumbel, and Weibull domains of attraction, with validation on simulated and real datasets.

Contribution

It introduces a method to construct confidence intervals around ME plots for tail behavior identification, extending previous results to Gumbel and Weibull domains.

Findings

Confidence bounds effectively distinguish tail types

Method validated on simulated data

Application demonstrated on real datasets

Abstract

In many practical situations exploratory plots are helpful in understanding tail behavior of sample data. The Mean Excess plot is often applied in practice to understand the right tail behavior of a data set. It is known that if the underlying distribution of a data sample is in the domain of attraction of a Frechet, Gumbel or Weibull distributions then the ME plot of the data tend to a straight line in an appropriate sense, with positive, zero or negative slopes respectively. In this paper we construct confidence intervals around the ME plots which assist us in ascertaining which particular maximum domain of attraction the data set comes from. We recall weak limit results for the Frechet domain of attraction, already obtained in Das and Ghosh (2013) and derive weak limits for the Gumbel and Weibull domains in order to construct confidence bounds. We test our methods on both simulated and real data sets.