Journal Article

Parameter Orthogonality for a Family of Discrete Distributions

Willmot, Gordon E.

1988

Overview

The standard contagious distributions (see Douglas 1980) have been used in such varied fields as biology and automobile insurance, often to model various physical phenomena as well as provide a good fit to count data when other models are inadequate. Unfortunately, the parameterizations often used when working with these distributions normally lead to extremely high correlations of the maximum likelihood estimators (MLE's). This tends to lead to mathematical complexities, and causes difficulty or even errors in their interpretation. Furthermore, numerical difficulties may arise when using numerical procedures to locate the estimates. Some of these difficulties were discussed by Douglas (1980, pp. 171, 204-205), who suggested that a reparameterization to reduce or even eliminate such correlation is desirable. If the MLE's are asymptotically uncorrelated, the parameterization is orthogonal. Philpot (1964) derived an orthogonal parameterization for the Neyman Type A distribution; Stein, Zucchini, and Juritz (1987) derived for the Poisson mixture by the inverse Gaussian distribution. Parameter orthogonality has several attractive features in the present context. Since there is no correlation asymptotically, the estimates (with their standard errors) provide a simpler summary of the data than in the absence of such orthogonality. The use of a parameterization where the MLEs are highly correlated can lead to a misleading analysis, or at best a more complicated analysis that would be necessary if an orthogonal parameterization had been used. To the extent that a high correlation exists, the parameters involved tend to measure similar quantities, and orthogonality separates information about the parameters from each other. This article gives an orthogonal parameterization for a large family of discrete distributions, including many of the contagious distributions, some Poisson mixtures, and some other generalized distributions. The previously cited works are unified and extended, for example, to the Polya-Aeppli, Poisson-binomial, and Sichel's Poisson-generalized inverse Gaussian distribution. One of the orthogonal parameters is the mean, and in many applications it is of interest to express the mean as a function of relevant covariate information. For example, Hinde (1982) considered some of these distributions in a regression context. This article shows how the results may be extended to deal with the covariate case in a relatively straightforward manner. Consequently, a convenient parameterization exists for a large family of distributions in a wide variety of situations. Some numerical examples are given, and a simple algorithm is given to find the maximum likelihood estimates in the case of no covariates.

• Publisher: Taylor & Francis Group,American Statistical Association
• Subject: Applications / Automobile insurance / Binomials / Contagious distribution / Correlations / Datasets / Exact sciences and technology / Mathematical independent variables / Mathematics / Maximum likelihood estimation / Maximum likelihood estimators / Mixed Poisson / Orthogonal parameterization / Orthogonality / Parameterization / Probability and statistics / Sciences and techniques of general use / Sichel distribution / Standard error / Statistics / Theory and Methods