This paper gives a proof that the likelihood ratio statistic, based on a sample x= (x₁,…,x_n) on a p-dimensional random variable X, converges in distribution to a noncentral chi-square distribution under a class of local alternatives, for a multi-dimensional parameter space. A proof of uniform convergence for this situation was 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. This paper is concerned with the case in which the information matrix is not of full rank. This generalizes the results of Silvey (1959), Davidson and Lever (1970) and EI-Helbawy and Soliman (1983).