The Two-Sample Problem Via Relative Belief Ratio

TL;DR

This paper introduces a Bayesian method for the two-sample problem using the relative belief ratio to compare distributions, demonstrating strong theoretical properties and excellent performance in examples.

Contribution

It proposes a novel Bayesian approach employing the relative belief ratio with Dirichlet process priors for testing equality of distributions.

Findings

The method effectively distinguishes between equal and different distributions.

Theoretical properties support the approach's validity.

Examples show excellent performance in various scenarios.

Abstract

This paper deals with a new Bayesian approach to the two-sample problem. More specifically, let $x=(x_1,\ldots,x_{n_1})$ and $y=(y_1,\ldots,y_{n_2})$ be two independent samples coming from unknown distributions $F$ and $G$, respectively. The goal is to test the null hypothesis $\mathcal{H}_0:~F=G$ against all possible alternatives. First, a Dirichlet process prior for $F$ and $G$ is considered. Then the change of their Cram\'{e}r-von Mises distance from a priori to a posteriori is compared through the relative belief ratio. Many theoretical properties of the procedure have been developed and several examples have been discussed, in which the proposed approach shows excellent performance.