Inferences under a stochastic ordering constraint: The k-sample case

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, . . .,Fk , 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 ˆ Fi and ˆ F ∗ i are the empirical and our restricted estimators of Fi , then we show that, asymptotically, P(| ˆ F ∗ i (x) − Fi (x)| ≤ u) ≥ P(| ˆ Fi (x) − Fi (x)| ≤ u) 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.