The asymptotic variance and skewness of maximum likelihood estimators using Maple †

In 1998, Bowman and Shenton introduced an asymptotic formula for the third central moment of a maximum likelihood estimator θ̂α of the parameter θα, a = 1, 2, . . . , s. From this moment, the asymptotic skewness can be set up using the standard deviation. Clearly, the skewness, measured in this way is location free, and scale free, so that shape is accounted for. The computer program is implemented by insertion of the values of expectations of products of logarithmic derivatives, a tiresome task. But now using Maple, the only input consists of the values of the parameters and the form of the density or probability function. Cases of up to four parameters have been implemented. However, in this paper we present twoand three-parameter cases in detail. Future improvements in handling Maple may lead to the implementation of the general case. Bowman and Shenton [Bowman, K.O. and Shenton, L.R., 1999, The asymptotic kurtosis for maximum likelihood estimators. Communications in Statistics, Theory and Methods, 28(11), 2641–2654.] also developed an asymptotic formula for the kurtosis, which is not used here. This study was initiated in our monograph [Shenton, L.R. and Bowman, K.O., 1977, Maximum Likelihood Estimation in Small Samples (Charles Griffin and Co., Ltd.)].