Asymptotic Relative Efficiency in Testing Asymptotic Relative Efficiency in Testing.Table 1 Some of efficiency cannot discriminate between them.On the local Bahadur efficiencies other hand,under some regularity conditions the one- Statistic Distribution sided tests like linear rank tests can be compared on the basis of their indices,and their Hodges-Lehmann effi- Hyperbolic ciency coincides locally with Bahadur efficiency,see details Gauss Logistic Laplace cosine Cauchy Gumbel in Nikitin (1995). 0 0.637 0.750 1 0.811 0.811 0.541 Coupled with three"basic"approaches to the ARE cal- culation described above,intermediate approaches are also 0.907 0.987 0.822 1 0.750 0.731 possible if the transition to the limit occurs simultane- ously for two parameters at a controlled way.Thus emerged Consider,for instance,the sample with the distribution the Chernoff ARE introduced by Chernoff(1952),see also function(df)F and suppose we are testing the goodness- Kallenberg (1982);the intermediate,or the Kallenberg of-fit hypothesis Ho F=Fo for some known continu- ARE introduced by Kallenberg(1983),and the Borovkov- ous df Fo against the alternative of location.Well-known Mogulskii ARE,proposed in Borovkov and Mogulskii distribution-free statistics for this hypothesis are the Kol- (1993). mogorov statistic D and omega-square statistic w.The Large deviation approach to asymptotic efficiency of following table presents their local absolute efficiency in tests was applied in recent years to more general prob- case of six standard underlying distributions: lems.For instance,the change-point,"signal plus white We see from Table 1 that the integral statistic is in noise"and regression problems were treated in Puhalskii most cases preferable with respect to the supremum-type and Spokoiny(1998),the tests for spectral density of a sta- statistic Dn.However,in the case of Laplace distribution tionary process were discussed in Kakizawa(2005),while the Kolmogorov statistic is locally optimal,the same hap- Taniguchi(2001)deals with the time series problems,and pens for the Cramer-von Mises statistic in the case of Otsu (2010)studies the empirical likelihood for testing hyperbolic cosine distribution.This observation can be moment condition models. explained in the framework of Bahadur local optimality, see Nikitin(1995 Chap.6). About the Author See also Nikitin (1995)for the calculation of local Bahadur efficiencies in case of many other statistics. Professor Nikitin is Chair of Probability and Statistics of St. Petersburg University.He is an Associate editor of Statistics Hodges-Lehmann efficiency and Probability Letters,and member of the editorial Board, This type of the ARE proposed in Hodges and Lehmann Mathematical Methods of Statistics and Metron.He is a (1956)is in the conformity with the classical Neyman- Fellow of the Institute of Mathematical Statistics.Profes Pearson approach.In contrast with Bahadur efficiency,let sor Nikitin is the author of the text Asymptotic efficiency us fix the level of tests and let compare the exponential rate of nonparametric tests,Cambridge University Press,NY, of decrease of their second-kind errors for the increasing 1995,and has authored more than 100 papers,in many number of observations and fixed alternative.This expo- international journals,in the field of Asymptotic efficiency nential rate for a sequence of statistics {T is measured of statistical tests,large deviations of test statistics and by some non-random function dr()which is called the nonparametric Statistics. Hodges-Lehmann index of the sequence {Tn).For two such sequences the Hodges-Lehmann ARE is equal to the Cross References ratio of corresponding indices. Asymptotic Relative Efficiency in Estimation The computation of Hodges-Lehmann indices is diffi- Chernoff-Savage Theorem cult as requires large deviation asymptotics of test statistics Nonparametric Statistical Inference under the alternative Robust Inference There exists an upper bound for the Hodges-Lehmann indices analogous to the upper bound for Bahadur exact slopes.As in the Bahadur theory the sequence of statistics References and Further Reading {Tn}is said to be asymptotically optimal in the Hodges- Arcones M(2005)Bahadur efficiency of the likelihood ratio test. Lehmann sense if this upper bound is attained. Math Method Stat 14:163-179 The drawback of Hodges-Lehmann efficiency is that Bahadur RR(1960)Stochastic comparison of tests.Ann Math Stat 31:276-295 most two-sided tests like Kolmogorov and Cramer-von Bahadur RR (1967)Rates of convergence of estimates and test Mises tests are asymptotically optimal,and hence this kind statistics.Ann Math Stat 38:303-324 A Asymptotic Relative Efficiency in Testing Asymptotic Relative Efficiency in Testing. Table  Some local Bahadur efficiencies Statistic Distribution Gauss Logistic Laplace Hyperbolic cosine Cauchy Gumbel Dn . .  . . . ω  n . . .  . . Consider, for instance, the sample with the distribution function (df) F and suppose we are testing the goodness￾of-t hypothesis H : F = F for some known continu￾ous df F against the alternative of location. Well-known distribution-free statistics for this hypothesis are the Kol￾mogorov statistic Dn and omega-square statistic ω  n. e following table presents their local absolute e›ciency in case of six standard underlying distributions: We see from Table  that the integral statistic ω  n is in most cases preferable with respect to the supremum-type statistic Dn. However, in the case of Laplace distribution the Kolmogorov statistic is locally optimal, the same hap￾pens for the Cramér-von Mises statistic in the case of hyperbolic cosine distribution. is observation can be explained in the framework of Bahadur local optimality, see Nikitin ( Chap. ). See also Nikitin () for the calculation of local Bahadur e›ciencies in case of many other statistics. Hodges–Lehmann efficiency is type of the ARE proposed in Hodges and Lehmann () is in the conformity with the classical Neyman￾Pearson approach. In contrast with Bahadur e›ciency, let us x the level of tests and let compare the exponential rate of decrease of their second-kind errors for the increasing number of observations and xed alternative. is expo￾nential rate for a sequence of statistics {Tn} is measured by some non-random function dT(θ) which is called the Hodges–Lehmann index of the sequence {Tn}. For two such sequences the Hodges–Lehmann ARE is equal to the ratio of corresponding indices. e computation of Hodges–Lehmann indices is di›- cult as requires large deviation asymptotics of test statistics under the alternative. ere exists an upper bound for the Hodges–Lehmann indices analogous to the upper bound for Bahadur exact slopes. As in the Bahadur theory the sequence of statistics {Tn} is said to be asymptotically optimal in the Hodges– Lehmann sense if this upper bound is attained. e drawback of Hodges–Lehmann e›ciency is that most two-sided tests like Kolmogorov and Cramér-von Mises tests are asymptotically optimal, and hence this kind of e›ciency cannot discriminate between them. On the other hand, under some regularity conditions the one￾sided tests like linear rank tests can be compared on the basis of their indices, and their Hodges–Lehmann e›- ciency coincides locally with Bahadur e›ciency, see details in Nikitin (). Coupled with three “basic” approaches to the ARE cal￾culation described above, intermediate approaches are also possible if the transition to the limit occurs simultane￾ously for two parameters at a controlled way.us emerged the Chernoš ARE introduced by Chernoš (), see also Kallenberg (); the intermediate, or the Kallenberg ARE introduced by Kallenberg (), and the Borovkov– Mogulskii ARE, proposed in Borovkov and Mogulskii (). Large deviation approach to asymptotic e›ciency of tests was applied in recent years to more general prob￾lems. For instance, the change-point, “signal plus white noise” and regression problems were treated in Puhalskii and Spokoiny (), the tests for spectral density of a sta￾tionary process were discussed in Kakizawa (), while Taniguchi () deals with the time series problems, and Otsu () studies the empirical likelihood for testing moment condition models. About the Author Professor Nikitin is Chair of Probability and Statistics of St. Petersburg University. He is an Associate editor of Statistics and Probability Letters, and member of the editorial Board, Mathematical Methods of Statistics and Metron. He is a Fellow of the Institute of Mathematical Statistics. Profes￾sor Nikitin is the author of the text Asymptotic e›ciency of nonparametric tests, Cambridge University Press, NY, , and has authored more than  papers, in many international journals, in the eld of Asymptotic e›ciency of statistical tests, large deviations of test statistics and nonparametric Statistics. Cross References 7Asymptotic Relative E›ciency in Estimation 7Chernoš-Savage eorem 7Nonparametric Statistical Inference 7Robust Inference References and Further Reading Arcones M () Bahadur efficiency of the likelihood ratio test. Math Method Stat :– Bahadur RR () Stochastic comparison of tests. Ann Math Stat :– Bahadur RR () Rates of convergence of estimates and test statistics. Ann Math Stat :–