On the asymptotic properties of a certain class of goodness-of-fit tests associated with multinomial distributions

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.

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

“…intermediate setting the known result that: if n  then the Pitman efficiency of the chi-square test wrt the d  -test is equal to 1. However, I conjecture that aforesaid property of PDS effects to IARE properties of h-tests if Nn .This is confirmed by the fact that the loglikelihood and chi-square tests have the same asymptotic efficiency in term of  -IAE for Nn ) , seeMirakhmedov (2021). But (ibid) -IARE of chi-square test is much inferior wrt tests satisfying the Cramer condition, for instance the log-likelihood and Freeman-Tukey tests, ifNn  and alternatives are such that IARE of the chi-square test wrt h-tests in the case Progress in the study of IARE h-tests depends on the progress of the results on the probability of large deviations of symmetric statistics (1.2).…”

“…In other words, we are interested in comparison of two h-tests in the situation, which are intermediate between Bahadur's and Pitman's concepts of asymptotic efficiency (AE), see Nikitin(1995), and hence the name. For the detailed comments on the alternatives (1.6) and brief survey we refer to Mirakhmedov (2021). In that work the comparison of h-tests was considered in terms of the so called " intermediate AE In the present paper we study the IARE of two h-tests, defined as a limit of the ratio of sample sizes which guarantee the same precision for both tests: the same significance level tending to 0 slower than exponentially and the same asymptotically non-degenerate power.…”

“…For the detailed comments on the alternatives (1.6) and brief survey we refer to Mirakhmedov (2021). In that work the comparison of h-tests was considered in terms of the so called "α − intermediate AE", according to which the performance of an h-test of size α measured by the asymptotic value of e α N (S h n ) = − log P 0 S h n > N A 1n (h) , provided that its power is asymptotically nondegenerate.…”