Valid asymptotic expansions for the likelihood ratio statistic and other perturbed chi-square variables

Abstract

Let {Z_n}_n≥1 be a sequence of random vectors. Under certain conditions, distributions of statistics which are smooth functions of the mean vector Z_n^- and whose asymptotic distributions are central Chi-square are shown to possess asymptotic expansions in powers of n^-1 As applications, asymptotic expansions of the null distributions of the likelihood ratio statistic, Wald's and Rao's statistics are obtained. The results proved here supplement the recent work of Bhattacharya and Ghosh (1978) and also justify the validity of the formal expansions obtained by Box (1949) and Hayakawa (1977).