Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions

Large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described. Specifically, the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived. These results generalize the work of Moran (1971), Chant (1974), and Chernoff (1954). Some of Chant's results are shown to be incorrect. The approach used in deriving these results follows from comments made by Moran and Chant. The problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate Gaussian distribution from a sample of size 1. In this representation the Gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator. Thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the Gaussian random variable onto the region of admissible values for the mean. A variety of examples is provided for which the limiting distributions of likelihood ratio statistics are mixtures of chi-squared distributions. One example is provided with a nuisance parameter on the boundary for which the asymptotic distribution is not a mixture of chi-squared distributions.