Chapter 8 - Asymptotic optimality and efficiency

Abstract

This chapter describes the asymptotic optimality and efficiency. The asymptotically optimum tests are presented. These tests are asymptotically optimum not only within the class of rank tests, but even within the class of all possible tests. Their optimality is preserved even for some subhypotheses, provided these subhypotheses also include the densities. It is also worth mentioning that in contrast to possible parametric solutions for testing the subhypotheses, the rank tests do not involve any estimation problems. It is found that a sequence of variate distribution functions converges completely for every continuous and uniformly bounded function. Complete information on the limiting proper ties of a test is provided by its asymptotic power. Comparing this asymptotic power with that of the asymptotically most powerful test, the asymptotic efficiency of the test is obtained. It is found that if all statistics under consideration have asymptotically normal distributions under both the hypothesis and the alternative, the notion of asymptotic efficiency may be defined by one number, independent of the level of significance and admitting a suggestive interpretation in sample problems.