Computational Statistics & Data Analysis 1 (1983) 239-255 North-Holland 239 On an integrated approach to member selection and parameter estimation for Pearson distributions Rudolph S. PARRISH Computer Sciences Corporation (c/o U.S. EPA, Athens, GA) Received February 1983 Revised September 1983 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 implemen- tation 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. Keywords and phrases: Estimation of distribution parameters, Loss functions, Pearson distributions. I. Introduction Whenever empirical data are to be summarized using a single distribution function, it is often convenient to fit a distribution which has been selected from an extensive family of empirical distributions. Two such families in common use are the Pearson system and the Johnson system [5]. Both of these consist of four-parameter distributions of various types. The Pearson system, for example, contains as members the normal, gamma, beta, and Student's t, among others. For both the Johnson system and the Pearson system, a member is selected for use in a given application usuaUy on the basis of sample estimates of the third and fourth standardized moments. Indeed, the selection procedure frequently is sum- marized geometrically by partitioning a subset of the skewness-kurtosis plane into regions corresponding to specific distribution types from the particular family (see Figure 1). However, as pointed out by Slifker and Shapiro [13], there are shortcomings to this procedure. Specifically they mention that the variances of 0167-9473/83/$3.00 © 1983, Elsevier Science Publishers B.V. (North-Holland)