Pearson's goodness-of-fit tests for sparse distributions

TL;DR

This paper investigates the behavior of Pearson's goodness-of-fit test in sparse, high-dimensional settings, showing convergence to normality and proposing a more powerful alternative, with real data application.

Contribution

It extends the theoretical understanding of Pearson's chi-squared test to sparse distributions and introduces a new, more powerful test statistic.

Findings

Chi-squared statistic converges to normal under general conditions

Proposed new test statistic outperforms chi-squared in simulations

Application demonstrated on lottery data

Abstract

Pearson's chi-squared test is widely used to test the goodness of fit between categorical data and a given discrete distribution function. When the number of sets of the categorical data, say $k$, is a fixed integer, Pearson's chi-squared test statistic converges in distribution to a chi-squared distribution with $k-1$ degrees of freedom when the sample size $n$ goes to infinity. In real applications, the number $k$ often changes with $n$ and may be even much larger than $n$. By using the martingale techniques, we prove that Pearson's chi-squared test statistic converges to the normal under quite general conditions. We also propose a new test statistic which is more powerful than chi-squared test statistic based on our simulation study. A real application to lottery data is provided to illustrate our methodology.