Inference for Likelihood Ratio Ordering in the Two-Sample Problem

Abstract

We obtain the maximum likelihood estimators of two multinomial probability vectors under the constraint that they are likelihood ratio ordered. We extend this estimation approach to the case of two univariate distributions and show strong consistency of the estimators. We also derive and study the asymptotic distribution of the likelihood ratio statistic for testing the equality of two discrete probability distributions against the alternative that one distribution is greater than the other in the likelihood ratio ordering sense. Finally, we examine a data set pertaining to average daily insulin dose from the Boston Collaborative Drug Surveillance Program and compare our testing procedure to testing procedures for other stochastic orderings.

Key Words:

• Chi-bar square distribution
• Hazard rate ordering
• Isotonic regression
• Multinomial distribution
• Stochastic ordering
• Strong consistency
• Uniform stochastic ordering