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Many times populations exist which logically must satisfy a stochastic ordering requirement. Nevertheless, estimates of these populations may not bear out this stochastic ordering because of the inherent variability of the observations. This paper will consider the problem of finding maximum likelihood estimates of stochastically ordered survival functions for the cases (a) one survival being fixed in advance and (b) estimating both survival functions when the data includes censored observations. A numerical example is handled in detail to illustrate the solution to this problem. (Author).
Very often, populations exist that, logically, should satisfy linear stochastic ordering requirements. For example if a mechanical device is improved through N stages, the corresponding survival functions should be linearly stochastically ordered. Nevertheless, estimates may not reflect this stochastic ordering because of the inherent variability of the observations. This document characterizes the maximum likelihood estimates of the survival functions subject to linear stochastic ordering requirements. These estimates may be expressed in terms of the well-known Kaplan-Meier product limit estimates. Also given is an iterative algorithm which must converge to the correct solution that depends only upon solving the pairwise problem. Finally the authors consider an example concerning survival times for people with squamous carcinoma in the oropharynx when classified by degree of lymph node deterioration at time of discovery. (Author).
A bibliography on stochastic orderings. Was there a real need for it? In a time of reference databases as the MathSci or the Science Citation Index or the Social Science Citation Index the answer seems to be negative. The reason we think that this bibliog raphy might be of some use stems from the frustration that we, as workers in the field, have often experienced by finding similar results being discovered and proved over and over in different journals of different disciplines with different levels of mathematical so phistication and accuracy and most of the times without cross references. Of course it would be very unfair to blame an economist, say, for not knowing a result in mathematical physics, or vice versa, especially when the problems and the languages are so far apart that it is often difficult to recognize the analogies even after further scrutiny. We hope that collecting the references on this topic, regardless of the area of application, will be of some help, at least to pinpoint the problem. We use the term stochastic ordering in a broad sense to denote any ordering relation on a space of probability measures. Questions that can be related to the idea of stochastic orderings are as old as probability itself. Think for instance of the problem of comparing two gambles in order to decide which one is more favorable.
The likelihood ratio principle is employed to suggest a nonparametric test for testing equality of two distributions against a stochastic ordering alternative. The test appears to be robust against a wide range of alternatives. Percentage points for sample sizes less than or equal to twenty are provided as well as a comparison of power values for the Kolmogorov-Smirnov and Mann-Whitney-Wilcoxon tests. (Author).