Comparison of Probability Distributions

Abstract

OFTEN, more than one probability distribution is theoretically feasible when considering statistical models for an experiment. The problem of determination of the more plausible distribution using likelihood procedures (see, for example, Sprott and Kalbfleisch, 1969) will be discussed for the simple case where all observations are made under the same response conditions. (Lindsey, 1974, will consider this problem when independent variables are present.) To do this using likelihood inference, a base statistical model must be introduced with which all other distributions under consideration may be compared. The derivation which follows yields the multinomial model as the base model. Several approaches have been suggested in the literature to the problem of determining which of a number of possible models best describes a set of data. Cox (1961, 1962) develops asymptotic Neyman-Pearson likelihood ratio tests and suggests an alternative approach involving a combination, either additive or multiplicative, of the density functions, with estimation of additional parameters. This approach is further developed by Atkinson (1970). When prior probabilities, both for each model and for the parameters within the models, are available, Lindley (1961, p. 456) gives a posterior odds ratio of the two models using Bayes's theorem. When applicable (i.e. when prior probabilities are available), this approach may be used with the methods developed below.