The Annals of Statistics

Statistical Inference for Uniform Stochastic Ordering in Several Populations

Richard Dykstra, Subhash Kochar, and Tim Robertson

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Stochastic ordering between probability distributions is a widely studied concept. It arises in numerous settings and has useful applications. Since it is often easy to make value judgments when such orderings exist, it is desirable to recognize their occurrence and to model distributional structure under such orderings. Unfortunately, the necessary theory for statistical inference procedures has not been developed for many problems involving stochastic ordering and this development seems to be a difficult task. We show in this paper that the stronger notion of uniform stochastic ordering (which is equivalent to failure rate ordering for continuous distributions) is quite tractable in matters of statistical inference. In particular, we consider nonparametric maximum likelihood estimation for $k$-population problems under uniform stochastic ordering restrictions. We derive closed-form estimates even with right-censored data by a reparameterization which reduces the problem to a well-known isotonic regression problem. We also derive the asymptotic distribution of the likelihood ratio statistic for testing equality of the $k$ populations against the uniform stochastic ordering restriction. This asymptotic distribution is of the chi-bar-square type as discussed by Robertson, Wright and Dykstra. These distributional results are obtained by appealing to elegant results from empirical process theory and showing that the proposed test is asymptotically distribution free. Recurrence formulas are derived for the weights of the chi-bar-square distribution for particular cases. The theory developed in this paper is illustrated by an example involving data for survival times for carcinoma of the oropharynx.

Article information

Ann. Statist., Volume 19, Number 2 (1991), 870-888.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing 62E20: Asymptotic distribution theory 62N05: Reliability and life testing [See also 90B25]

Stochastic ordering uniform stochastic ordering isotonic regression empirical processes Kaplan-Meier estimates level probabilities least squares projections failure rate ordering chi-bar-square distribution


Dykstra, Richard; Kochar, Subhash; Robertson, Tim. Statistical Inference for Uniform Stochastic Ordering in Several Populations. Ann. Statist. 19 (1991), no. 2, 870--888. doi:10.1214/aos/1176348125.

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