On Obtaining Large-Sample Tests from Asymptotically Normal Estimators

Abstract

This is an extension of Wald's asymptotic test procedure based on unrestricted maximum-likelihood estimators. Wald showed that under certain regularity conditions the test statistic has a limiting central chi-square distribution under the hypothesis and a limiting noncentral chi-square distribution under a sequence of local alternatives. We extend this procedure, allowing it to be based on a broader class of estimators and to obey simpler and less restrictive conditions. Sufficient conditions for validity of the limiting distributions are local twice-differentiability of the left side of the hypothesis and, under a sequence of local alternatives, asymptotic normality of the estimator of the parameter defining the distribution and stochastic convergence (to the appropriate asymptotic value) of the estimator of the covariance matrix. The required asymptotic behavior is verified for the case of independent sampling from two normal distributions and formulas are presented which aid in computing the test statistic.