Valid asymptotic expansions for the likelihood ratio and other statistics under contiguous alternatives

Abstract

Let {Z_n}_n≥1 be a sequence of i.i.d. random vectors. Let W_n be a statistic based on the mean vector Z_n^- whose asymptotic null distribution is a central chi-square with p degrees of freedom. It is shown that the distribution function under contiguous alternatives of W_n possesses a valid asymptotic expansion in powers of n^-½ , the leading term being a noncentral chi-square with p degrees of freedom and the coefficients of n^-j/2(j≥ 0) being finite linear combinations of noncentral chi-squares with same noncentrality parameter and with degrees of freedom p,p+2,p+4,...., provided the conditions of Chandra and Ghosh (1979) together with a uniform Cramer's condition and smoothness conditions on moments hold. The result is applied to get expansions for the likelihood ratio statistic, Wald's and Rao's statistics under contiguous alternatives. The similar expansion for the likelihood ratio statistic obtained formally by Hayakawa (1977) has been justified.