Probabilities for asymmetric p-outside values

TL;DR

This paper introduces new asymmetric p-outside value functions and box-plots that better account for distribution tail asymmetry, aiding in tail analysis and parameter estimation.

Contribution

It proposes novel theoretical and empirical methods for analyzing distribution tails with asymmetry, improving upon symmetric quantile-based approaches.

Findings

New asymmetric p-outside value functions introduced

Functions are independent of distribution center and scale

Examples demonstrate applicability to tail comparison and parameter estimation

Abstract

In 2017-2020 Jordanova and co-authors investigate probabilities for p-outside values and determine them in many particular cases. They show that these probabilities are closely related to the concept for heavy tails. Tukey's boxplots are very popular and useful in practice. Analogously to the chi-square-criterion, the relative frequencies of the events an observation to fall in different their parts, compared with the corresponding probabilities an observation of a fixed probability distribution to fall in the same parts, help the practitioners to find the accurate probability distribution of the observed random variable. These open the door to work with the distribution sensitive estimators which in many cases are more accurate, especially for small sample investigations. All these methods, however, suffer from the disadvantage that they use inter quantile range in a symmetric way. The concept for outside values should take into account the form of the distribution. Therefore, here, we give possibility for more asymmetry in analysis of the tails of the distributions. We suggest new theoretical and empirical box-plots and characteristics of the tails of the distributions. These are theoretical asymmetric p-outside values functions. We partially investigate some of their properties and give some examples. It turns out that they do not depend on the center and the scaling factor of the distribution. Therefore, they are very appropriate for comparison of the tails of the distribution, and later on, for estimation of the parameters, which govern the tail behaviour of the cumulative distribution function.