A Unified Approach to Likelihood Inference on Stochastic Orderings in a Nonparametric Context

Abstract

For data in a two-way contingency table with ordered margins, we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities. This leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log-likelihood ratio tests. The usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases. In particular, for each of these three orderings we determine the transformation required to apply the estimation algorithm; we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered, as well as testing each stochastic order against an unrestricted alternative. We show that in all cases the test statistics are asymptotically distributed as a mixture of chi-squared distributions, with weights determined by the information matrix. By exploiting the special structure of this matrix in these three cases, we find tight upper and lower bounds to the distribution of all test statistics. These bounding distributions are free of nuisance parameters and relatively easy to compute. Two examples are presented to illustrate the methodology and the required computations needed to apply these techniques.

Key Words:

• Chi-bar-squared distribution
• Likelihood inference
• Likelihood ratio ordering
• Order-restricted inference
• Stochastic ordering
• Uniform ordering.