Asymptotic normality and efficiency for certain goodness-of-fit tests

Abstract

SUMMARY Assume that a random sample of size n has been taken from a multinomial distribution with N cells. Let ξ k be the number of observations in the k th cell and set Z n =(ξ 1 , 1/ N )+ F n (ξ n , N / N ). It is proved that, under general conditions, Z n is asymptotically normal when n and N tend to infinity so that n / N ↑ α > 0. This result is used to study asymptotic efficiency for certain goodness-of-fit tests. The chi-squared statistic turns out to be optimal in some situations.