Abstract: The object of this paper is to describe the development
of ideas 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 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 the sample size) term in the third central moment of was found, and a year later a
rather complicated expression for the 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 and biases, the and variances, the third central moment, and the 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 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.
