Parameter Estimation for the Sichel Distribution and Its Multivariate Extension

Abstract

Abstract The Sichel distribution is a three-parameter compound Poisson distribution. It is a versatile model for highly skewed frequency distributions of observed counts and has proved useful in fields as diverse as mining engineering, linguistics, ecology, industrial psychology, and market research. We propose a reparameterization of the Sichel distribution and give an algorithm, which can be implemented on a typical desktop microcomputer, for computing the maximum likelihood estimates of the new parameters. The reparameterization has a number of advantages over the old. In the important two-parameter special case of the Sichel distribution known as the inverse Gaussian Poisson the new parameters are the population mean and a shape parameter, and their maximum likelihood estimators are asymptotically uncorrelated. The reparameterization also lends itself to the convenient multivariate extension presented here. This distribution is well suited for modeling correlated count data whose marginal distributions exhibit the long sparse tails characteristic of the univariate Sichel distribution. Properties and maximum likelihood estimators of this multivariate Sichel distribution are considered. Examples of application for both univariate and bivariate cases are given. Since the Sichel distribution encompasses a number of the well-known discrete distributions as limiting forms (Sichel 1971), the estimates of the parameters sometimes suggest an appropriate limiting form for the data. This is illustrated in one of the examples.