Marginal distributions of maximum likelihood estimator when one or two components of the true parameter are on the boundary of the parameter space

When the true parameter lies on the boundary of the parameter space it is difficult to investigate the asymptotic distribution of maximum likelihood estimator. In some relatively simple cases it is a mixture of truncated normal distributions. In this paper we shall be concerned with the marginal distributions of maximum likelihood estimator when one or two components of the true parameter are zero and can be on the boundary of the parameter space. We found that these distributions are (mixtures of) normal or truncated normal multiplied by “skew functions” which distort the symmetry of the normality. Some of these are skew¬normal.