The object of this paper is to describe the development of ideas of pertaining to sample size and maximum likelihood estimators of parameters associated with a probability function or density function. About forty years ago we considered a Taylor type series for a maximum likelihood estimator for qa, there being s parameters. First order bias and first order variance were included. Because of limitations in computer facilities, the skewness and kurtosis were avoided, and also because of the complicated structures involved. But toward the end of the 20th century an expression for the N-2 (N the sample size) term in the third central moment of was found, and a year later a rather complicated expression for the N-3 term in the fourth central moment was discovered. The skewness and kurtosis expressions involved much heavier work in deriving expectations of products of log-derivatives of the probability function or density, especially when 3 or more parameters were involved. At this stage we used the Maple symbolic language to cope with the N-1 and N-2 biases, the N-1 and N-2 variances, the N-2 third central moment, and the N-3 fourth central moment. We use the to measure skewness. This ratio is location and scale free, and it takes into account the shape of the distribution involved. Since under normality, we can set the observed value for a parameter qa, to a small value e and deduce a safe – sample size to achieve pseudo normality. Programs are provided in detail for the low order moments of a maximum likelihood estimator, simultaneous estimation being involved.
Bias, Variance, Skewness and Kurtosis of Maximum Likelihood Estimators Using Maple*