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

Assume that a random sample of size n has been taken from a multinomial distribution with N cells. Let ξk be the number of observations in the kth cell and set Zn=(ξ1, 1/N)+Fn(ξn,N/N). It is proved that, under general conditions, Zn 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.