On an integrated approach to member selection and parameter estimation for Pearson distributions

Abstract

Empirical data often are summarized by selecting and fitting an appropriate member from a wide class of frequency distributions. A common technique used for selection of a density function and estimation of its parameters relies upon the method of moments. In the case of Pearson distributions, it is possible to adopt a loss function approach and thereby avoid some of the problems associated with the moments method. Discussed here is a general technique that can be utilized for selecting and fitting, concurrently, a Pearson distribution on the basis of any of several loss functions. Except for Pearson distribution evaluation routines, which are available, only commonly available numerical minimization methods are required. In order to facilitate implementation of this procedure, analytical derivatives of Pearson density functions are provided. In addition, the technique is illustrated for cases of maximum likelihood, minimum chi-square, truncated Pearson distributions, estimation of Pearson priors, and an application from chemistry.