Tests For Stochastic Ordering Research Articles

Estimating a distribution function subject to a stochastic order restriction: a comparative study

In this article, we compare four nonparametric estimators of a distribution function (DF), estimated under a stochastic order restriction. The estimators are compared by simulation using four criteria: (1) the estimation of cumulative DFs; (2) the estimation of quantiles; (3) the estimation of moments and other functionals; and (4) as tools for testing for stochastic order. Our simulation study shows that estimators based on the pointwise maximum-likelihood estimator (p-MLE) outperform all other estimators when the underlying distributions are ‘close’ to each other. The gain in efficiency may be as high as 25%. If the DFs are far apart then the p-MLE may not be the best. However, the efficiency loss using the p-MLE relative to the best estimator in each case is generally low (about 5%). We also find that the test based on the p-MLE is the most powerful in the majority of cases although the gain in power relative to other tests is generally small.

Nonparametric tests for stochastic ordering

We present two new tests for stochastic ordering in a standard two-sample scheme. We approach the problem via its reparametrization in terms of Fourier coefficients in some corresponding system of functions and combining the resulting empirical Fourier coefficients. The empirical Fourier coefficients can be seen to be the asymptotically optimal linear rank statistics for the local sequences of nonparametric alternatives related to the introduced system of functions. Therefore, our first construction of the test is via multiple testing. The second test is based on sum of squares of censored empirical Fourier coefficients with the number of summands determined via a new model selection rule. The selection rule is fully automatic. Extensive simulations show that the new solutions improve upon existing tests based on adjusted variants of classical Kolmogorov–Smirnov, Anderson–Darling and L 1-distance-based statistics, among others. We show that both tests control the error of the first kind for any fixed sample sizes and are capable of detecting essentially any alternative as the sample sizes are growing to infinity. We also discuss several aspects of our constructions, including possible efficiency calculations and asymptotic comparisons.

A numerical comparison of the normal and some saddlepoint approximations to a distribution-free test for stochastic ordering in the competing risks model

Calculating the exact critical value of the test statistic is important in nonparametric statistics. However, to evaluate the exact critical value is difficult when the sample sizes are moderate to large. Under these circumstances, to consider more accurate approximation for the distribution function of a test statistic is extremely important. A distribution-free test for stochastic ordering in the competing risks model has been proposed by Bagai et al. (1989). Herein, we performed a saddlepoint approximation in the upper tails for the Bagai statistic under finite sample sizes. We then compared the saddlepoint approximations with the Bagai approximation and investigate the accuracy of the approximations. Additionally, the orders of errors of the saddlepoint approximations were derived.

Conditional inference under simultaneous stochastic ordering constraints

Testing for stochastic ordering is of considerable importance when increasing does of a treatment are being compared, but in applications involving multivariate responses has received much less attention. We propose a permutation test for testing against multivariate stochastic ordering. This test is distribution-free and no assumption is made about the dependence relations among variables. A comparative simulation study shows that the proposed solution exhibits a good overall performance when compared with existing tests that can be used for the same problem.

The bias of linear rank tests when testing for stochastic order in ordered categorical data

In many hypothesis testing problems, the alternative hypothesis is characterized by one or several restrictions arising from a natural ordering among the outcome levels. It is known that tests which ignore the ordering may lack adequate statistical power for the alternatives which are of the most interest. Considerably less, however, is known about developing tests which conform to a natural ordering. Stochastic order is an objective and compelling characterization of the superiority of one treatment over another. Consequently, testing for stochastic order is of considerable importance in applications involving the comparison of one treatment to another on the basis of ordered categorical data (e.g., in clinical trials). With few exceptions (that we identify), there is no optimal test for this problem, so it is reasonable to consider the class of tests that are simultaneously unbiased and admissible. There are known necessary and sufficient conditions for admissibility, but unbiasedness is more elusive. It is known that a necessary condition for unbiasedness is exactness conditionally on the margins. Further, one can construct a test which is both admissible and unbiased by repeatedly improving the trivially unbiased “ignore-the-data” test. This complicated approach, however, does not, in general, lead to a nested family of tests. We provide a new necessary condition for unbiasedness, and show that linear rank tests, which are widely used and locally most powerful, fail this condition for certain sets of margins. For such margins, linear rank tests are severely biased (with power as low as zero to detect certain alternatives of interest) and least stringent.

Convex Hull Test for Ordered Categorical Data

When testing for stochastic order in ordered 2 x J contingency tables, it is common to select the cutoff required to declare significance so as to ensure that the size of the test is exactly alpha conditionally on the margins. It is valid, however, to use the margins to select not only the cutoff but also the form of the test. Linear rank tests, which are locally most powerful and frequently used in practice, suffer from the drawback that they may have power as low as zero to detect some alternatives of interest when the margins satisfy certain conditions. The Smirnov and convex hull tests are shown, through exact conditional power calculations and simulations, to avoid this drawback. The convex hull test is also admissible and palindromic invariant and minimizes the required significance level to have limiting power of one as the alternative moves away from the null in any direction.

A Likelihood Ratio Test for Stochastic Ordering

Abstract This article presents a method of extending the likelihood ratio principle to a nonparametric setting. The problem dealt with is a test for stochastic ordering. The resulting test is shown, via simulation, to have good power. It is shown to be consistent and unbiased.

A Likelihood Ratio Test for Stochastic Ordering

Abstract This article presents a method of extending the likelihood ratio principle to a nonparametric setting. The problem dealt with is a test for stochastic ordering. The resulting test is shown, via simulation, to have good power. It is shown to be consistent and unbiased.