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This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest , , , based on the pseudolikelihood , where is a consistent estimator of , the nuisance parameter. We show that the asymptotic distribution of T under H_{0} is a weighted sum of independent chi-squared variables. Some sufficient conditions are provided for the limiting distribution to be a chi-squared variable. When the true value of the parameter of interest, , or the true value of the nuisance parameter, , lies on the boundary of parameter space, the problem is shown to be asymptotically equivalent to the problem of testing the restricted mean of a multivariate normal distribution based on one observation from a multivariate normal distribution with misspecified covariance matrix, or from a mixture of multivariate normal distributions. A variety of examples are provided for which the limiting distributions of T may be mixtures of chi-squared variables. We conducted simulation studies to examine the performance of the likelihood ratio test statistics in variance component models and teratological experiments.