Topics Involving the Gamma Distribution: the Normal Coefficient of Variation and Conditional Monte Carlo.

Abstract

A transformation of the sample coefficient of variation ($CV$) for normal data is shown to be nearly proportional to a $chiˆ2$ random variable. The associated density is applied to inference on the common $CV$ of $k$ populations, testing $CV$ homogeneity across populations, and confidence intervals for the ratio of two $CV$s. The resulting tests and confidence intervals are shown via theory and simulation to be valid and powerful.

In other work on the coefficient of variation, a sample of scientific abstracts is used to characterize the values of the $CV$ encountered in practice, point estimation for a common $CV$ in normal populations is studied, and the literature on testing $CV$ homogeneity in normal populations is reviewed.

There is very little literature on the problem of conducting inference in models for continuous data conditional on sufficient statistics for nuisance parameters. This thesis explores Monte Carlo approaches to conditional $p$-value calculation in such models, including Dirichlet data generation, importance sampling, Markov chain Monte Carlo, and a method related to fiducial inference. Importance sampling is used to create a conditional test of $CV$ homogeneity in normal populations using the $chi^2$ approximation mentioned above. A Markov chain Monte Carlo solution is given to the long-standing problem of testing the homogeneity of exponential populations subject to Type I censoring. Conditional Monte Carlo algorithms are also applied to testing for an effect of a factor in an experiment with exponential data, testing for a dispersion effect in a replicated experiment with normal data, and testing a null value of a coefficient in exponential regression with an inverse link; brief consideration is also given to the problem of testing the homogeneity of $k$ $gamma$ distributions.