On the maximum likelihood ratio method of deciding between the weibull and gamma distributions

Abstract

The aims of this paper are to develop a procedure for finding the sample size required for deciding between two overlapping families of distributions: Weibull and gamma, and to evaluate the asymptotic behavior of the maximum likelihood estimators of percentiles of these distributions. We base our procedure on the logarithm of the ratio of the maximized likelihoods. An introduction to this statistic was given in Cox (1961, 1962) and it was applied to this problem by Bain and Englehardt (1980). We show how to find the sample size needed for a user specified probability of correct selection using a normal approximation to the distribution of the Cox statistic.

The consequences of making an incorrect choice of family could result in erroneous estimates of true percentiles, particularly when using maximum likelihood estimators. We discuss the effects of these errors.

Keywords:

• Gamma distributions versus Weibull distributions
• Weibull distributions versus gamma distributions
• Reliability
• Likelihood ratio tests
• Selection procedure
• Separate families of hypotheses