Asymptotic Efficiency of Nonparametric Tests

Abstract

Making a substantiated choice of the most efficient statistical test is one of the basic problems of statistics. Asymptotic efficiency is an indispensable technique for comparing and ordering statistical tests in large samples. It is especially useful in nonparametric statistics where it is usually necessary to rely on heuristic tests. This monograph presents a unified treatment of the analysis and calculation of the asymptotic efficiencies of nonparametric tests. Powerful new methods are developed to evaluate explicitly different kinds of efficiencies. Of particular interest is the description of domains of the Bahadur local optimality and related characterisation problems based on recent research by the author. Other Russian results are also published here for the first time in English. Researchers, professionals and students in statistics will find this book invaluable.

Vanishing shortcoming and asymptotic relative efficiency

“…Corollary 5.9). This result supplies analogous results stated in another framework by Neuhaus (1976) and Nikitin (1995). Moreover, the asymptotic relative efficiency under other alternatives than the one given by C 1 (x) will be calculated.…”

“…The proof of Lemma 5.1 follows from inequalities in Inglot and Ledwina (1990). The constant π involved in (5.4) can be deduced from Nikitin (1995), e.g., while a related result on the tails of the limiting distribution of T n can be found in Gregory (1980), cf. also Theorem 2.1 and Proposition 2.3 in .…”

“…They provide a simple and intuitive way of comparison of quadratic tests with the best possible one or of two quadratic tests with each other. They supply Nikitin's (1995) results, where Bahadur's approach has been exploited. On the other hand, they can be nicely confronted with the two step approach proposed by Hájek and Sidák (1967) and applied in Neuhaus (1976) as well as Wieand's approach exploited by Gregory (1980).…”

A New Method for Comparing Experiments and Measuring Information

“…In In studying different types of efficiencies in the most are functions of the sample size (see Nikitin (1995) for a survey of asymptotic (1954, p. 227) refers to sequences X n which consist of increasingly informative observations. suppose our experiment consists of taking ten observations, X x x x ( , , , , ) K .…”

On the Bahadur slope of the Lilliefors and the Cramér–von Mises tests of normality

“…For a review on Bahadur asymptotic optimality see and Nikitin [16]. The Bahadur slopes of the tests in (1.1) and (1.2) can be found in Chapter 2 in [16]. For the statistic in (1.1), it is known (see [6]) that if F 0 is a continuous c.d.f., then…”