## All Graduate Theses and Dissertations

5-1971

Thesis

#### Degree Name:

Master of Science (MS)

#### Department:

Mathematics and Statistics

#### Department name when degree awarded

Applied Statistics

Ronald V. Canfield

#### Abstract

Four families of contagious distributions--generalized Poisson distributions, generalized binomial distributions, generalized Pascal distributions, and generalized log-zero distributions--are investigated in this thesis.

The family of generalized Poisson distributions contains five distributions: the Neyman Type A, the "Short," the Poisson binomial, the Poisson Pascal, and the negative binomial. The family of generalized binomial distributions contains eight distributions: the binomial Poisson, the binomial binomial, the binomial Pascal, the binomial log-zero, the Poisson with zeros, the binomial with zeros, the Pascal with zeros, and the log-zero with zeros. The family of generalized Pascal distributions contains four distributions: the Pascal Poisson, the Pascal binomial, the Pascal Pascal, and the Pascal log-zero. The family of generalized log-zero distributions contains four distributions: the log-zero Poisson, the log-zero binomial, the log-zero Pascal, and the log-zero log-zero.

For each family of contagious distributions, the common probability generating function based on a biological model is derived by application of Feller's compound distribution theorem and Gurland's generalized distribution terminology. The common recurrence relation and the common factorial moments or cumulants are derived from the common probability generating function by using the successive differentiation method. Then for each distribution within this family, the particular probability generating function, recurrence relation, and factorial moments or cumulants are easily obtained from common ones. The equations of factorial moments or cumulants are solved.

The maximum likelihood equations are derived for some distributions which have been shown to provide a good or excellent moment fitting. These equations are solved by an iteration procedure, Except for the Neyman Type A distribution and the "Short" distribution in which the maximum likelihood equations are derived from the probability generating functions and solved by the method of scoring, the maximum likelihood equations a re derived from the probability functions and solved by the Newton-Raphson method.

Forty sets of biological and accident data classified into five types have been collected from various sources. A Fortran program has been written for fitting each distribution and a numerical example is given to illustrate the fitting procedure.

In comparing the fits among these distributions, the chi-square goodness- of-fit values have been calculated and tabulated. The results suggest that the binomial distribution with zeros and the Pascal distribution with zeros be used if one is to describe the empirical data arising from populations having a contagious character. This is not only due to the fact that the two distributions have provided better fits to all five types of data, but also the fact that their maximum likelihood estimate procedures have no common disadvantages of other distributions. These disadvantages are that not every moment estimate can allow the iteration process to converge and that the probabilities must be recalculated after each iteration.

#### Checksum

8b4f96138a59a65d319694a2db516d35

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