On the intermediate asymptotic efficiency of goodness-of-fit tests in multinomial distributions

TL;DR

This paper investigates the intermediate asymptotic efficiency of goodness-of-fit tests for multinomial distributions, focusing on power divergence statistics and comparing chi-square with other symmetric tests as the number of cells grows with sample size.

Contribution

It provides a detailed analysis of the intermediate asymptotic relative efficiency of chi-square and other symmetric tests in multinomial goodness-of-fit testing.

Findings

Chi-square test's efficiency is characterized relative to other symmetric tests.

The study covers asymptotic regimes where the number of cells increases with sample size.

Results inform the choice of tests under intermediate asymptotic conditions.

Abstract

We consider goodness-of-fit tests for uniformity of a multinomial distribution by means of tests based on a class of symmetric statistics, defined as the sum of some function of cell-frequencies. We are dealing with an asymptotic regime, where the number of cells grows with the sample size. Most attention is focused on the class of power divergence statistics. The aim of this article is to study the intermediate asymptotic relative efficiency of two tests, where the powers of the tests are asymptotically non-degenerate and the sequences of alternatives converge to the hypothesis, but not too fast. The intermediate asymptotic relative efficiency of the chi-square test wrt an arbitrary symmetric test is considered in details.