Conditional Goodness-of-Fit Tests for Discrete Distributions

TL;DR

This paper introduces new likelihood-based goodness-of-fit tests for discrete distributions, especially the geometric distribution, using a conditional testing approach that allows for exact p-value calculation and compares favorably with classical tests.

Contribution

It develops a novel conditional testing framework for discrete distributions, including methods for exact p-value computation and simulation, extending to various distributions.

Findings

New tests outperform classical methods in simulations

Conditional sampling method enables exact p-value calculation

Framework adaptable to multiple discrete distributions

Abstract

In this paper, we address the problem of testing goodness-of-fit for discrete distributions, where we focus on the geometric distribution. We define new likelihood-based goodness-of-fit tests using the beta-geometric distribution and the type I discrete Weibull distribution as alternative distributions. The tests are compared in a simulation study, where also the classical goodness-of-fit tests are considered for comparison. Throughout the paper we consider conditional testing given a minimal sufficient statistic under the null hypothesis, which enables the calculation of exact p-values. For this purpose, a new method is developed for drawing conditional samples from the geometric distribution and the negative binomial distribution. We also explain briefly how the conditional approach can be modified for the binomial, negative binomial and Poisson distributions. It is finally noted that the simulation method may be extended to other discrete distributions having the same sufficient statistic, by using the Metropolis-Hastings algorithm.