Asymptotic skewness and the distribution of maximum likelihood estimators

The maximum likelihood procedure to estimate parameters of a model has several attractive properties including the existence of the covariance matrix which yields asymptotic covariances; for a sample size N the asymptotics are in general of order 1/N. Here we give an asymptotic for the skewness of the distribution of the maximum likelihood estimator of a parameter; this is of order 1/N-2 and this expression is new. Applications relate to the parameters of (i) the Poisson, binomial, and normal density, (ii) the gamma, density and (iii) the Beta density. Other applications are being considered. The expression for the asymptotic skewness at one phase of I-he study turned out to be unusually complicated involving the asymptotic expressions for variance and bias. When these were identified a much simpler compact expression appeal-ed which we now describe. The work is a. much improved treatment of the subject described in Shenton and Bowman (Maximum likelihood estimation in small samples, Griffin, 1977).