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

• 成果类型: Article
• 署名作者: El Barmi, H; Mukerjee, H
• 署名单位: City University of New York (CUNY) System; Baruch College (CUNY); Wichita State University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1198/016214504000000764
• 发表日期: 2005
• 页码: 252-261
• 关键词: maximum-likelihood-estimation; survival functions; nonparametric-estimation; random-variables; isotonic regression; ratio test; distributions; populations; tests

摘要

If X-1 and X-2 are random variables with distribution functions F-1 and F-2, then X-1 is said to be stochastically larger than X-2 if F-1 <= F-2. Statistical inferences under stochastic ordering for the two-sample case has a long and rich history. In this article we consider the k-saniple case; that is, we have k populations with distribution functions F-1, F-2,.... F-k, k >= 2, and we assume that F-1 <= F-2 <= (...) <= F-k. For k = 2, the nonparametric maximum likelihood estimators of F, 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 (F) over cap (i) and (F) over cap (i)* are the empirical and our restricted estimators of F-i, then we show that, asymptotically, P(vertical bar(F) over cap (i)*(x) - F-i(x)vertical bar <= u) >= P(vertical bar(F) over cap (i)(x) - F-i(x)vertical bar <= 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.