Characterizations of non-normalized discrete probability distributions and their application in statistics

TL;DR

This paper develops explicit formulas for non-normalized discrete distributions using Stein's method, enabling statistical analysis without normalization constants, and demonstrates applications in distribution fitting and parameter estimation.

Contribution

It introduces new identities for non-normalized discrete distributions and applies them to statistical problems like distribution testing and parameter estimation.

Findings

Effective methods for testing Poisson fit

Parameter estimation for negative binomial distributions

Application to non-normalized exponential-polynomial models

Abstract

From the distributional characterizations that lie at the heart of Stein's method we derive explicit formulae for the mass functions of discrete probability laws that identify those distributions. These identities are applied to develop tools for the solution of statistical problems. Our characterizations, and hence the applications built on them, do not require any knowledge about normalization constants of the probability laws. To demonstrate that our statistical methods are sound, we provide comparative simulation studies for the testing of fit to the Poisson distribution and for parameter estimation of the negative binomial family when both parameters are unknown. We also consider the problem of parameter estimation for discrete exponential-polynomial models which generally are non-normalized.