THE LIMITING DISTRIBUTION OF THE LIKELIHOOD RATIO STATISTIC UNDER A CLASS OF LOCAL ALTERNATIVES

The paper gives a proof that -21 n lambda sub n, the likelihood ratio statistic based on a sample of size n, converges in distribution to a noncentral chi-square distribution under local alternatives to the null hypothesis for a multi-dimensional parameter space. Consideration is limited to maximum likelihood estimates that are solutions to the likelihood equations obtained for the maximization process. Proof of uniform convergence for this situation has been given by Wald (Wald, A. (1943) Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Amer. Math. Soc. 54, 426-482.), whose assumptions include the uniform consistency of the maximum likelihood estimates and of the likelihood ratio test. The assumptions utilized in this paper can be more directly verified in applications than those required by Wald. (Author)