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 ₙ denote the maximum likelihood estimator, and consider the case where pₙ tends to infinity with n and where \ₙ\ is a sequence of parameter values in R^pₙ. Moment conditions are provided under which \|ₙ - ₙ\| = Oₚ (pₙ/n) and \|ₙ - ₙ - Xₙ\| = Oₚ (pₙ/n), where Xₙ is the sample mean. The latter result provides normal approximation results when p²ₙ/n 0. It is shown by example that even for a single coordinate of (ₙ - ₙ), p²ₙ/n 0 may be needed for normal approximation. However, if p^3/2ₙ/n 0, the likelihood ratio test statistic for a simple hypothesis has a chi-square approximation in the sense that (-2 - pₙ) /2pₙ D N (0, 1).