Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

A Note on the Moment Generating Function

A simple proof is given to show that there always exists a neighborhood of zero in which a moment generating function has a power series expansion. Thus, the relation between moments and derivatives of the moment generating function at zero can be obtained without resorting to postcalculus theorems.     

The Teacher's Corner: Interpolation in F-Tables

A lemma is given which simplifies the derivation of maximum likelihood estimates of the covariance matrix in the multivariate normal case.     

The Teacher's Corner: An Interactive Computer Quizzing System to Support the Proctorial System of Instruction

Although the importance of statistics in many phases of university activities is well-known, many universities have not given serious thought to the optimum organizational structure for statistics. This article addresses this problem in two parts: 1. Principles and/or goals of optimum administrative structures for statistics, and 2. Discussion of the most popular organizational structures now in existence together with comments on the advantages and disadvantages of each. The article concludes that a central statistics unit (department) appears the most optimal structure, although it is not without certain disadvantages and/or dangers.     

Testing the Fit of Gamma Distributions Using the Empirical Moment Generating Function

ABSTRACT

This article presents goodness-of-fit tests for two and three-parameter gamma distributions that are based on minimum quadratic forms of standardized logarithmic differences of values of the moment generating function and its empirical counterpart. The test statistics can be computed without reliance to special functions and have asymptotic chi-squared distributions. Monte Carlo simulations are used to compare the proposed test for the two-parameter gamma distribution with goodness-of-fit tests employing empirical distribution function or spacing statistics. Two data sets are used to illustrate the various tests.