Noncentral Convergence of Wald's Large-Sample Test Statistic in Exponential Families

Abstract

It is shown that under very mild assumptions Wald's large-sample test statistic (quadratic form based on unrestricted maximum likelihood estimators) converges to noncentral chi-square under a sequence of local alternatives of the order $n^{-\frac{1}{2}}$, when the family of distributions is assumed to be of exponential type. This eliminates, for these families, the necessity of invoking the strict regularity conditions of Wald for the purpose of justifying the asymptotic distribution.