An Extension of a Truncated Poisson Distribution

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Abstract
In order to provide a model for the distribution of certain randomly distributed organisms which may be observed at randomly selected sites, the truncated Poisson distribution with missing zero class is extended by adding a selection parameter. The resulting probability function is written as where [image] [lambda] is the Poisson parameter and is the proportion of potential sites that fall in the "selected" category. Based on a sample consisting of a total of N observations of random variable x, the maximum likelihood estimator of [theta] is shown to be [theta]= n[image]/N where n[image] is the number of nonzero sample observations. The maximum likelihood estimator of [lambda] is shown to be the same as the estimator of this parameter based on a total sample of n[image] observations from an ordinary truncated Poisson distribution when the zero class is missing as considered by the same author in Estimating the parameter in a conditional Poisson distribution (Biometrics 16(2):203-211. 1960). Asymptotic variances of these estimators are given and the asymptotic covariance is shown to be zero. An illustrative example, consisting of observed frequencies of Pyrausta nubilalis in 1937 was selected from Beall, Geoffrey and Rescia, Richard R., A generalization of Neyman''s contagious distributions (Biometrics 9: 354-86. 1953). The extended Poisson distribution considered here provides a better fit to these data than any of the various contagious distributions employed by Beall and Rescia.