Magnitude Orders

A stochastic order is a partial order that quantifies the concept of one random variable being bigger than (or more variable or more skewed) another in some probabilistic sense. In this chapter various types of stochastic orders which compare the magnitudes of two random variables and their properties are discussed. These include the (usual) stochastic order, the hazard rate (also called the uniform stochastic) order, and the likelihood ratio order. Also, the stochastic orders for comparing the components of a random vector are discussed where concepts like the joint likelihood ratio, the joint hazard rate and the joint stochastic orders are described. The problem of paired data is also considered where one would like to compare the control variable with the treatment variable. The concept of the hazard rate order is extended to the cause-specific hazard rate order. Applications of the P-P plots and the P-P order are also discussed. Several examples are also given to explain these concepts. Some techniques for developing nonparametric tests for stochastic and hazard rate orders are described. We also discuss the problem of testing for the equality of two distributions against the alternative that they cross at a single point as well as estimating the point of crossing. A brief discussion on testing for the joint likelihood ratio ordering is also given. The emphasis is on motivation and ideas for developing new statistical tests and estimators rather than on data analysis.