Discrete convolution statistic for hypothesis testing

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 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.

Keywords:

• Discrete convolution
• sum of discrete random variables
• statistical hypothesis testing
• maximum-likelihood estimation
• sub-independence

Mathematics Subject Classification:

• 62G05
• 62G10
• 62G20
• 62P10
• 62P20

Code availability

Python 3 code for simulation and testing is publicly available at “https://github.com/ GiulioPr/Discrete_convolution_statistic.”

Acknowledgments

We thank Julia Marchingo, Andrey Kan, Susanne Heinzel and Phil Hodgkin at the Walter and Eliza Hall Institute of Medical Research for collaborating with us on the scientific problem that motivated the development of the convolution statistic and associated tests.

Additional information

Funding

The research leading to these results has received funding from the European Union Seventh Framework Program (FP7/2007–2013) under grant agreement 317040 (QuanTI) and by Science Foundation Ireland Grant 12IP1263.