A Non-parametric Approach to Inference about the Tail of a Continuous or a Discrete Distribution

TL;DR

This paper presents a non-parametric, information-theoretic method for identifying and estimating the tail types of both discrete and continuous distributions, aiding model selection and analysis.

Contribution

It introduces the concept of tail profile and a plotting-based approach to classify distribution tails, extending to continuous data via binning under certain conditions.

Findings

Effective in distinguishing tail types including exponential, near-exponential, sub-exponential, and power-law.

Provides point and interval estimates for parameters of thicker-than-exponential tails.

Demonstrates good performance through simulations across various distribution scenarios.

Abstract

This article introduces a non-parametric information-theoretic approach to inference about the tail of a continuous or a discrete distribution. Leveraging a new concept named tail profile -- a set of information-theoretic quantities developed from results of domains of attraction on countable alphabets -- theoretical evidence supports the identification of specific discrete distributional tail types through a sequence of plots. The approach discerns tail types by bench-marking against exponential, and three thicker-than-exponential families: near-exponential, sub-exponential, and power-law (zipf, Pareto). For tails thicker-than-exponential, the approach also provides point and interval estimates for some of the underlying distribution parameters. While primarily designed to streamline the selection of discrete parametric models for detailed statistical analysis, a supporting theorem enables the method's extension use to continuous data, stating that binning continuous data with a common width preserves the tail decay rate under certain conditions. Simulations are presented to demonstrate the method's performance across various scenarios.