Tails and probabilities for extreme outliers

TL;DR

This paper advances methods for estimating the tails of probability distributions with small samples by analyzing new tail characteristics and proposing six distribution-sensitive estimators, comparing their performance with existing estimators.

Contribution

It introduces new properties of tail heaviness characteristics and develops six novel estimators for the extremal index, enhancing tail analysis in small samples.

Findings

New properties of tail heaviness characteristics are established.

Six new estimators for the extremal index are proposed.

Simulation results compare these estimators with existing methods.

Abstract

The task of estimation of the tails of probability distributions having small samples seems to be still opened and almost unsolvable. The paper tries to make a step in filling this gap. In 2017 Jordanova et al. introduce six new characteristics of the heaviness of the tails of theoretical distributions. They rely on the probability to observe {\color{blue}mild or} extreme outliers. The main their advantage is that they always exist. This work presents some new properties of these characteristics. Using them six distribution sensitive estimators of the extremal index are defined. A brief simulation study compares their quality with the quality of Hill, t-Hill, Pickands and Deckers-Einmahl-de Haan estimators.