Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity

Consider a sample of size n from a regular exponential family in Pn dimensions. Let 6, denote the maximum likelihood estimator, and consider the case where Pn tends to infinity with n and where {(On} is a sequence of parameter values in RP-. Moment conditions are provided under which II6 0,JI = O?( p(,/n) and On6n On Xnll = Op (p,/n), where Xn is the sample mean. The latter result provides normal approximation results when p2/n 0. It is shown by example that even for a single coordinate of (6, in), pn2/n -? 0 may be needed for normal approximation. However, if pn3/2 /n O 0, the likelihood ratio test statistic A for a simple hypothesis has a chi-square approximation in the sense that (2 log A pn )/ 2Pn D A(0, 1).