Discrete convolution statistic for hypothesis testing

TL;DR

This paper introduces a new convolution-based statistic for hypothesis testing of equality in distribution between two sums of discrete independent variables, improving power especially with small samples.

Contribution

It proposes the convolution statistic as a nonparametric maximum likelihood estimator and derives its asymptotic distribution for hypothesis testing, connecting it with probability generating functions.

Findings

Convolution test outperforms Pearson's χ² in simulations.

The method effectively uses all data, enhancing power in small samples.

Guidelines for applying the convolution test are provided.

Abstract

The question of testing for equality in distribution between two linear models, each consisting of sums of distinct discrete independent random variables with unequal numbers of observations, has emerged from the biological research. In this case, the computation of classical $\chi^2$ statistics, which would not include all observations, results in loss of power, especially when sample sizes are small. Here, as an alternative that uses all data, the nonparametric maximum likelihood estimator for the distribution of sum of discrete and independent random variables, which we call the convolution statistic, is proposed and its limiting normal covariance matrix determined. To challenge null hypotheses about the distribution of this sum, the generalized Wald's method is applied to define a testing statistic whose distribution is asymptotic to a $\chi^2$ with as many degrees of freedom as the rank of such covariance matrix. Rank analysis also reveals a connection with the roots of the probability generating functions associated to the addend variables of the linear models. A simulation study is performed to compare the convolution test with Pearson's $\chi^2$, and to provide usage guidelines.