On the Asymptotic Properties of a Certain Class of Goodness-of-Fit Tests Associated with Multinomial Distributions

TL;DR

This paper investigates the asymptotic behavior of goodness-of-fit tests for multinomial distributions, focusing on power divergence statistics like chi-square and log-likelihood ratio, under intermediate asymptotic efficiency conditions.

Contribution

It provides a detailed analysis of the asymptotic properties of these tests within the intermediate efficiency framework, extending understanding of their performance under converging alternatives.

Findings

Asymptotic properties of tests are characterized under intermediate efficiency.

Chi-square and log-likelihood ratio tests are analyzed in this framework.

Results show bounded power for certain alternative sequences.

Abstract

The object of study is the problem of testing for uniformity of the multinomial distribution. We consider tests based on symmetric statistics, defined as the sum of some function of cell-frequencies. Mainly, attention is focused on the class of power divergence statistics, in particular, on the chi-square and log-likelihood ratio statistics. The main issue of the article is to study the asymptotic properties of tests at the concept of an intermediate setting in terms of so called -intermediate asymptotic efficiency due to Ivchenko and Mirakhmedov (1995), when the asymptotic power of tests are bounded away from zero and one, while sequences of alternatives converge to the hypothesis, but not too fast.