The Limiting Distribution of the Likelihood Ratio Statistic under a Class of Local Alternatives

This paper gives a proof that$-2\,{\rm ln}\lambda _{n}$, 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. A proof of uniform convergence for this situation has been given by Wald (1943) 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.