A Bayesian Study of the Multinomial Distribution

摘要

Lindley (Ann.Math.Stat. 1964) studied the topic in the title. By using Fisher's conditional- Poisson approach to the multinomial and the logarithmic transformation of gamma variables to normality, he showed heuristically that linear contrasts in the logarithms of the cell probabilities theta sub i are asymptotically jointly normal and suggested that the approximation can be improved by applying a 'correction' to the sample. By studying the asymptotic series for the joint distribution, we have verified this assertion and found an improved correction procedure. A more detailed expansion is given for the distribution of a single contrast. In many problems linear functions of the theta sub i are of interest. The exact distribution for these is obtained. This has a density of a form familiar in the theory of serial correlation coefficients. A beta approximation is given. For three cells, a numerical example is given to show the merit of this approximation. The exact joint distribution of two linear functions of the theta sub i is also obtained and the theory is applied to a genetic linkage example. (Author)