Fitting tails affected by truncation

TL;DR

This paper develops a unified method for analyzing truncated tail data in extreme value analysis, accounting for physical and measurement truncation effects across various applications.

Contribution

It generalizes the Peaks over Threshold method to include truncation effects for heavy and light tails, using a pseudo-maximum likelihood approach.

Findings

Effective estimation of tail parameters with truncation considered

Simulation results support the proposed method

Asymptotic properties derived for the estimators

Abstract

In several applications, ultimately at the largest data, truncation effects can be observed when analysing tail characteristics of statistical distributions. In some cases truncation effects are forecasted through physical models such as the Gutenberg-Richter relation in geophysics, while at other instances the nature of the measurement process itself may cause under recovery of large values, for instance due to flooding in river discharge readings. Recently Beirlant et al. (2016) discussed tail fitting for truncated Pareto-type distributions. Using examples from earthquake analysis, hydrology and diamond valuation we demonstrate the need for a unified treatment of extreme value analysis for truncated heavy and light tails. We generalise the classical Peaks over Threshold approach for the different max-domains of attraction with shape parameter $\xi>-1/2$ to allow for truncation effects. We use a pseudo-maximum likelihood approach to estimate the model parameters and consider extreme quantile estimation and reconstruction of quantile levels before truncation whenever appropriate. We report on some simulation experiments and provide some basic asymptotic results.