Statistical Tests For Stochastic Orders

Stochastic ordering is a partial ordering relation, by which random variables are compared according to their probability distributions. Research on stochastic ordering can be traced back to the 1930’s, in which Hardy Littlewood and Po’lya proposed a dominance relation of the two nonnegative vectors. A large number of statisticians then studied the comparison of random variables especially since Lehmann introduced the concept of stochastic ordering in 1955. From the 60’s to the early 70’s, the researches on this topic have been extended rapidly, and then widely used to various practical problems, such as the bioinformatics, reliability, queuing theory, insurance mathematics, risk management and other fields.With more and more extensive applications of stochastic ordering, the theory has also been fast developed, and many types of stochastic orderings have been proposed in order to accommodate the applications in different areas. For example, simple stochastic ordering, increasing convex (concave) ordering, likelihood ratio ordering, failure rate ordering, Lorenz order and dispersion ordering are defined. At the same time, statistical inference of stochastic ordering also has attracted the attention of many scholars, and a lot of research results have been obtained. But most of these studies are conducted for two populations, and the methods for more than two populations are required in practice. This thesis will discuss statistical tests of simple stochastic ordering, increasing concave ordering, likelihood ratio ordering for multiple populations. More specifically, the research contents of this thesis are summarized as follows:1. Statistical test of simple stochastic ordering within multiple populations. We consider the null hypothesis which is the stochastic equality of k(>2) distribu-tions, and the alternative hypothesis of the stochastic monotonicity within k(>2) distributions under simple stochastic ordering. For the study of the test, firstly, we construct the test statistics by the isotonic regression estimations, considering that the isotonic regression estimations can ensure the estimations of distribution functions ordered. The obtained test statistics can reflect the order relation of distribution functions and enhance the power of test. Secondly, under some con-ditions, we give the asymptotic null distribution of the test statistic. However, the resulted distribution is very complicated, and depends on the underlying un-known distributions, thus it is difficult to be used directly to compute the critical value. Therefore, we give the critical value by bootstrap method. Finally, through simulation results we show that the proposed methodology performs well.2. Statistical test of increasing concave ordering within multiple populations. We consider the null hypothesis of the stochastic equality of k(>2) distributions, and the alternative hypothesis of the stochastic monotonicity under increasing con-cave ordering. Two methods are proposed to meet different applications. The first method uses the iterative method which was proposed by Hogg in 1962. Firstly, we decompose the test problem into a sequence of sub hypotheses, each sub hy-pothesis test can be regarded as a test for two populations, which is easier to deal with. Then we construct the test statistics by the sequence of sub test statistics. Finally, we obtain the critical value by an inequality. The second method employs the isotonic regression estimations and bootstrap method. Firstly, we give the iso-tonic regression estimations of the integral of the distribution functions. Then we construct K-S statistics by the isotonic regression estimations, and give the limit distribution of the test statistic under some conditions. However, the limit distri-bution is very complicated, and depends on the underlying unknown distributions, thus it is difficult to be used directly to compute the critical value. Finally, we show the performance of the proposed methods and compare their advantages and disadvantages through simulations.3. Statistical test of likelihood ratio ordering within multiple populations. The null hypothesis we considered is the stochastic monotonicity of distributions under likelihood ratio ordering, and the alternative hypothesis is no restriction. We discuss the problem by the method of the likelihood ratio test. Firstly, we construct the test statistic by the likelihood function. Then we prove that the asymptotic distribution of the test statistic is the weighted chi squared distribution. Finally, through simulation results we show that the proposed method performs well.At the end of the thesis, we extend the test of likelihood ratio ordering under more than two populations to the test of nonlinear inequality constraints, and obtain similar theoretical results to that for likelihood ratio ordering. Simulation results are also presented to show that the proposed method performs well.