The Method of Maximum Likelihood Applied to the Poisson Binomal Distribution

Much has been written about the application of contagious distributions to plant and insect populations, for a discussion of the methods and a review of the literature see McGuire, Brindley, and Bancroft [1957]. Three distributions considered in the past are the negative binomial, the Neyman Type A, and the Poisson binomial. The method of maximum likelihood has been used to fit the negative binomial, as described by Fisher and Bliss [1953], and to fit the Neyman Type A, as described by Douglas [1955], and by Shenton [1949]. It is the purpose of this paper to describe a procedure for fitting the Poisson binomial distribution by the method of maximum likelihood, and to consider the efficiencies of the method of moments and the method of sample zero frequency. Since the Neyman Type A distribution is a limiting form of the Poisson binomial, the formulae developed here will include as special cases some of the formulae developed previously for the Neyman Type A.