Inferences Under a Stochastic Ordering Constraint

Abstract

If X₁ and X₂ are random variables with distribution functions F₁ and F₂, then X₁ is said to be stochastically larger than X₂ if F₁ ≤F₂. Statistical inferences under stochastic ordering for the two-sample case has a long and rich history. In this article we consider the k-sample case; that is, we have k populations with distribution functions F₁, F₂, … , F_k,k ≥ 2, and we assume that F₁ ≤ F₂ ≤ ˙˙˙ ≤ Fk. For k = 2, the nonparametric maximum likelihood estimators of F₁ and F₂ under this order restriction have been known for a long time; their asymptotic distributions have been derived only recently. These results have very complicated forms and are hard to deal with when making statistical inferences. We provide simple estimators when k ≥ 2. These are strongly uniformly consistent, and their asymptotic distributions have simple forms. If and are the empirical and our restricted estimators of F_i, then we show that, asymptotically, for all x and all u > 0, with strict inequality in some cases. This clearly shows a uniform improvement of the restricted estimator over the unrestricted one. We consider simultaneous confidence bands and a test of hypothesis of homogeneity against the stochastic ordering of the k distributions. The results have also been extended to the case of censored observations. Examples of application to real life data are provided.