2 Asymptotic Relative Efficiency in Testing Properties of Estimators efficiency of tests is more complicated than that of asymp- Statistical Fallacies:Misconceptions,and Myths totic efficiency of estimates.Various approaches to this notion were identified only in late forties and early fifties, References and Further Reading hence,20-25 years later than in the estimation theory.We Basu D(1956)On the concept of asymptotic relative efficiency. proceed now to their description. Sankhya 17:193-196 Let {Tn}and [Vn}be two sequences of statistics Hodges JL,Lehmann EL(1963)Estimates of location based on rank based on n observations and assigned for testing the null- tests.Ann Math Stat 34:598-611 Lehmann EL (1998a)Elements of large-sample theory.Springer, hypothesis H against the alternative A.We assume that New York the alternative is characterized by real parameter and Lehmann EL(1998b)Nonparametrics:statistical methods based on fore=0o turns into H.Denote by Nr(a,β，θ)the sam- ranks.Prentice-Hall,Upper Saddle River,NJ ple size necessary for the sequence [Tn}in order to attain Lehmann EL,Casella G(1988)Theory of point estimation,2nd edn. the power B under the level a and the alternative value Springer,New York Monohan JF(1984)Algorithm 616:fast computation of the Hodges- of parameter.The number Nv(ca,β，θ)is defined in the Lehmann location estimator.ACM T Math Software 10:265-270 same way. Nikitin Y(1995)Asymptotic efficiency of nonparametric tests.Cam- It is natural to prefer the sequence with smaller N. bridge University Press,Cambridge Therefore the relative efficiency of the sequence [Tn}with Nikitin Y(2010)Asymptotic relative efficiency in testing.Interna- respect to the sequence [Vn}is specified as the quantity tional Encyclopedia of Statistical Sciences.Springer,New York Puhalskii A,Spokoiny V(1998)On large-deviation efficiency in statistical inference.Bernoulli 4:203-272 er.v(aβ，θ)=Nv(a,β，θ)/Nr(a,f,θ) Rousseeuw PI,Croux C(1993)Alternatives to the median absolute deviation.J Am Stat Assoc 88:1273-1283 Serfling R(1980)Approximation Theorems of Mathematical Statis- so that it is the reciprocal ratio of sample sizes Nr and Ny. tics.Wiley,New York Serfling R(2002)Efficient and robust fitting of lognormal distribu- The merits of the relative efficiency as means for tions.N Am Actuarial J 4:95-109 comparing the tests are universally acknowledged.Unfor- Serfling R,Wackerly DD(1976)Asymptotic theory of sequential tunately it is extremely difficult to explicitly compute fixed-width confidence interval procedures.J Am Stat Assoc Nr(ag,β，θ)even for the simplest sequences of statistics 71:949-955 [Tn.At present it is recognized that there is a possi- Staudte RG.Sheather SJ(1990)Robust estimation and testing.Wiley, New York bility to avoid this difficulty by calculating the limiting values er.v(a,β，θ)as0→fo,asa→0 and as B1 keeping two other parameters fixed.These lim- iting values er.v,e.v and e are called respectively Asymptotic Relative Efficiency the Pitman,Bahadur and Hodges-Lehmann asymptotic in Testing relative efficiency(ARE),they were proposed correspond- ingly in Pitman(1949),Bahadur(1960)and Hodges and YAKOV NIKITIN Lehmann (1956). Professor,Chair of Probability and Statistics Only close alternatives,high powers and small levels St.Petersburg University,St.Petersburg,Russia are of the most interest from the practical point of view.It keeps one assured that the knowledge of these ARE types will facilitate comparing concurrent tests,thus producing Asymptotic Relative Efficiency of Two well-founded application recommendations. Tests The calculation of the mentioned three basic types Making a substantiated choice of the most efficient statis- of efficiency is not easy,see the description of theory tical test of several ones being at the disposal of the statis- and many examples in Serfling (1980),Nikitin (1995) tician is regarded as one of the basic problems of Statistics. and Van der Vaart (1998).We only mention here,that This problem became especially important in the middle Pitman efficiency is based on the central limit theorem of XX century when appeared computationally simple but (see Central Limit Theorems)for test statistics.On the “inefficient”rank tests. contrary,Bahadur efficiency requires the large deviation Asymptotic relative efficiency(ARE)is a notion which asymptotics of test statistics under the null-hypothesis, enables to implement in large samples the quantitative while Hodges-Lehmann efficiency is connected with large comparison of two different tests used for testing of the deviation asymptotics under the alternative.Each type of same statistical hypothesis.The notion of the asymptotic efficiency has its own merits and drawbacks
A Asymptotic Relative Efficiency in Testing 7Properties of Estimators 7Statistical Fallacies: Misconceptions, and Myths References and Further Reading Basu D () On the concept of asymptotic relative efficiency. Sankhya :– ¯ Hodges JL, Lehmann EL () Estimates of location based on rank tests. Ann Math Stat :– Lehmann EL (a) Elements of large-sample theory. Springer, New York Lehmann EL (b) Nonparametrics: statistical methods based on ranks. Prentice-Hall, Upper Saddle River, NJ Lehmann EL, Casella G () Theory of point estimation, nd edn. Springer, New York Monohan JF () Algorithm : fast computation of the Hodges– Lehmann location estimator. ACM T Math Software :– Nikitin Y () Asymptotic efficiency of nonparametric tests. Cambridge University Press, Cambridge Nikitin Y () Asymptotic relative efficiency in testing. International Encyclopedia of Statistical Sciences. Springer, New York Puhalskii A, Spokoiny V () On large-deviation efficiency in statistical inference. Bernoulli :– Rousseeuw PJ, Croux C () Alternatives to the median absolute deviation. J Am Stat Assoc :– Serfling R () Approximation Theorems of Mathematical Statistics. Wiley, New York Serfling R () Efficient and robust fitting of lognormal distributions. N Am Actuarial J :– Serfling R, Wackerly DD () Asymptotic theory of sequential fixed-width confidence interval procedures. J Am Stat Assoc :– Staudte RG, Sheather SJ () Robust estimation and testing. Wiley, New York Asymptotic Relative Efficiency in Testing Yakov Nikitin Professor, Chair of Probability and Statistics St. Petersburg University, St. Petersburg, Russia Asymptotic Relative Efficiency of Two Tests Making a substantiated choice of the most ecient statistical test of several ones being at the disposal of the statistician is regarded as one of the basic problems of Statistics. is problem became especially important in the middle of XX century when appeared computationally simple but “inecient” rank tests. Asymptotic relative eciency (ARE) is a notion which enables to implement in large samples the quantitative comparison of two dierent tests used for testing of the same statistical hypothesis. e notion of the asymptotic eciency of tests is more complicated than that of asymptotic eciency of estimates. Various approaches to this notion were identied only in late forties and early ies, hence, – years later than in the estimation theory. We proceed now to their description. Let {Tn} and {Vn} be two sequences of statistics based on n observations and assigned for testing the nullhypothesis H against the alternative A. We assume that the alternative is characterized by real parameter θ and for θ = θ turns into H. Denote by NT(α, β, θ) the sample size necessary for the sequence {Tn} in order to attain the power β under the level α and the alternative value of parameter θ. e number NV(α, β, θ) is dened in the same way. It is natural to prefer the sequence with smaller N. erefore the relative eciency of the sequence {Tn} with respect to the sequence {Vn} is specied as the quantity eT,V(α, β, θ) = NV(α, β, θ)/NT(α, β, θ) , so that it is the reciprocal ratio of sample sizes NT and NV. e merits of the relative eciency as means for comparing the tests are universally acknowledged. Unfortunately it is extremely dicult to explicitly compute NT(α, β, θ) even for the simplest sequences of statistics {Tn}. At present it is recognized that there is a possibility to avoid this diculty by calculating the limiting values eT,V(α, β, θ) as θ → θ, as α → and as β → keeping two other parameters xed. ese limiting values e P T,V, e B T,V and e HL T,V are called respectively the Pitman, Bahadur and Hodges–Lehmann asymptotic relative eciency (ARE), they were proposed correspondingly in Pitman (), Bahadur () and Hodges and Lehmann (). Only close alternatives, high powers and small levels are of the most interest from the practical point of view. It keeps one assured that the knowledge of these ARE types will facilitate comparing concurrent tests, thus producing well-founded application recommendations. e calculation of the mentioned three basic types of eciency is not easy, see the description of theory and many examples in Sering (), Nikitin () and Van der Vaart (). We only mention here, that Pitman eciency is based on the central limit theorem (see 7Central Limit eorems) for test statistics. On the contrary, Bahadur eciency requires the large deviation asymptotics of test statistics under the null-hypothesis, while Hodges–Lehmann eciency is connected with large deviation asymptotics under the alternative. Each type of eciency has its own merits and drawbacks
Asymptotic Relative Efficiency in Testing Pitman Efficiency If the condition of asymptotic normality fails,consid- A Pitman efficiency is the classical notion used most often for erable difficulties arise when calculating the Pitman ARE the asymptotic comparison of various tests.Under some as the latter may not at all exist or may depend on a regularity conditions assuming asymptotic normality of and B.Usually one considers limiting Pitman ARE as test statistics under H and A,it is a number which has been a0.Wieand(1976)has established the correspondence gradually calculated for numerous pairs of tests. between this kind of ARE and the limiting approximate We quote now as an example one of the first Pitman's Bahadur efficiency which is easy to calculate. results that stimulated the development of nonparametric statistics.Consider the two-sample problem when under Bahadur Efficiency the null-hypothesis both samples have the same continu- The Bahadur approach proposed in Bahadur(1960;1967) ous distribution and under the alternative differ only in to measuring the ARE prescribes one to fix the power location.Let ew.be the Pitman ARE of the two-sample of tests and to compare the exponential rate of decrease Wilcoxon rank sum test (see Wilcoxon-Mann-Whitney of their sizes for the increasing number of observa- Test)with respect to the corresponding Student test(see tions and fixed alternative.This exponential rate for a Student's t-Tests).Pitman proved that for Gaussian sam- sequence of statistics [T}is usually proportional to some ples ew.=3/m0.955,and it shows that the ARE of non-random function cr()depending on the alternative the Wilcoxon test in the comparison with the Student test parameter 0 which is called the exact slope of the sequence (being optimal in this problem)is unexpectedly high.Later [Tn).The Bahadur ARE er(0)of two sequences Hodges and Lehmann(1956)proved that of statistics [Vn)and {Tn}is defined by means of the formula 0.864≤eiw,t≤+o∞， ev.r(0)=cv(0)/cr(0). if one rejects the assumption of normality and,moreover, It is known that for the calculation of exact slopes it is nec- the lower bound is attained at the density essary to determine the large deviation asymptotics of a sequence T under the null-hypothesis.This problem 3(5-x2)/(20v5) ifx≤V5, is always nontrivial,and the calculation of Bahadur effi- f(x)= 0 otherwise. ciency heavily depends on advancements in large deviation theory,see Dembo and Zeitouni(1998)and Deuschel and Hence the Wilcoxon rank test can be infinitely better Stroock (1989). than the parametric test of Student but their ARE never It is important to note that there exists an upper bound falls below 0.864.See analogous results in Serfling (2010) for exact slopes cr(8)≤2K(8) where the calculation of ARE of related estimators is discussed. in terms of Kullback-Leibler information number K() Another example is the comparison of independence which measures the "statistical distance"between the tests based on Spearman and Pearson correlation coef- alternative and the null-hypothesis.It is sometimes com- ficients in bivariate normal samples.Then the value of pared in the literature with the Cramer-Rao inequality in Pitman efficiency is 9/n20.912. the estimation theory.Therefore the absolute(nonrelative) In numerical comparisons,the Pitman efficiency Bahadur efficiency of the sequence {Tn}can be defined as appear to be more relevant for moderate sample sizes than e(8)=cr(θ)/2K(θ). other efficiencies Groeneboom and Oosterhoff(1981).On It is proved that under some regularity conditions the other hand,Pitman ARE can be insufficient for the the likelihood ratio statistic is asymptotically optimal in comparison of tests.Suppose,for instance,that we have Bahadur sense(Bahadur 1967;Van der Vaart 1998,Sect. a normally distributed sample with the mean 6 and vari- 16.6;Arcones2005). ance 1 and we are testing H:0=0 against A:0>0.Let Often the exact Bahadur ARE is uncomputable for any compare two significance tests based on the sample mean alternative 0 but it is possible to calculate the limit of X and the Student ratio t.As the t-test does not use the Bahadur ARE as approaches the null-hypothesis.Then information on the known variance,it should be inferior one speaks about the local Bahadur efficiency. to the optimal test using the sample mean.However,from The indisputable merit of Bahadur efficiency consists the point of view of Pitman efficiency,these two tests are in that it can be calculated for statistics with non-normal equivalent.On the contrary,Bahadur efficiency e()is asymptotic distribution such as Kolmogorov-Smirnov, strictly less than 1 for any 00. omega-square,Watson and many other statistics
Asymptotic Relative Efficiency in Testing A A Pitman Efficiency Pitman eciency is the classical notion used most oen for the asymptotic comparison of various tests. Under some regularity conditions assuming 7asymptotic normality of test statistics under H and A, it is a number which has been gradually calculated for numerous pairs of tests. We quote now as an example one of the rst Pitman’s results that stimulated the development of nonparametric statistics. Consider the two-sample problem when under the null-hypothesis both samples have the same continuous distribution and under the alternative dier only in location. Let e P W, t be the Pitman ARE of the two-sample Wilcoxon rank sum test (see 7Wilcoxon–Mann–Whitney Test) with respect to the corresponding Student test (see 7Student’s t-Tests). Pitman proved that for Gaussian samples e P W, t = /π ≈ . , and it shows that the ARE of the Wilcoxon test in the comparison with the Student test (being optimal in this problem) is unexpectedly high. Later Hodges and Lehmann () proved that . ≤ e P W, t ≤ +∞, if one rejects the assumption of normality and, moreover, the lower bound is attained at the density f (x) = ⎧⎪⎪ ⎨ ⎪⎪⎩ ( − x )/ (√ ) if ∣x∣ ≤ √ , otherwise. Hence the Wilcoxon rank test can be innitely better than the parametric test of Student but their ARE never falls below .. See analogous results in Sering () where the calculation of ARE of related estimators is discussed. Another example is the comparison of independence tests based on Spearman and Pearson correlation coef- cients in bivariate normal samples. en the value of Pitman eciency is /π ≈ .. In numerical comparisons, the Pitman eciency appear to be more relevant for moderate sample sizes than other eciencies Groeneboom and Oosterho (). On the other hand, Pitman ARE can be insucient for the comparison of tests. Suppose, for instance, that we have a normally distributed sample with the mean θ and variance and we are testing H : θ = against A : θ > . Let compare two signicance tests based on the sample mean X¯ and the Student ratio t. As the t-test does not use the information on the known variance, it should be inferior to the optimal test using the sample mean. However, from the point of view of Pitman eciency, these two tests are equivalent. On the contrary, Bahadur eciency e B t,X¯ (θ) is strictly less than for any θ > . If the condition of asymptotic normality fails, considerable diculties arise when calculating the Pitman ARE as the latter may not at all exist or may depend on α and β. Usually one considers limiting Pitman ARE as α → . Wieand () has established the correspondence between this kind of ARE and the limiting approximate Bahadur eciency which is easy to calculate. Bahadur Efficiency e Bahadur approach proposed in Bahadur (; ) to measuring the ARE prescribes one to x the power of tests and to compare the exponential rate of decrease of their sizes for the increasing number of observations and xed alternative. is exponential rate for a sequence of statistics {Tn} is usually proportional to some non-random function cT(θ) depending on the alternative parameter θ which is called the exact slope of the sequence {Tn}. e Bahadur ARE e B V,T(θ) of two sequences of statistics {Vn} and {Tn} is dened by means of the formula e B V,T(θ) = cV(θ) / cT(θ). It is known that for the calculation of exact slopes it is necessary to determine the large deviation asymptotics of a sequence {Tn} under the null-hypothesis. is problem is always nontrivial, and the calculation of Bahadur e- ciency heavily depends on advancements in large deviation theory, see Dembo and Zeitouni () and Deuschel and Stroock (). It is important to note that there exists an upper bound for exact slopes cT(θ) ≤ K(θ) in terms of Kullback–Leibler information number K(θ) which measures the “statistical distance” between the alternative and the null-hypothesis. It is sometimes compared in the literature with the 7Cramér–Rao inequality in the estimation theory. erefore the absolute (nonrelative) Bahadur eciency of the sequence {Tn} can be dened as e B T(θ) = cT(θ)/K(θ). It is proved that under some regularity conditions the likelihood ratio statistic is asymptotically optimal in Bahadur sense (Bahadur ; Van der Vaart , Sect. .; Arcones ). Oen the exact Bahadur ARE is uncomputable for any alternative θ but it is possible to calculate the limit of Bahadur ARE as θ approaches the null-hypothesis. en one speaks about the local Bahadur eciency. e indisputable merit of Bahadur eciency consists in that it can be calculated for statistics with non-normal asymptotic distribution such as Kolmogorov-Smirnov, omega-square, Watson and many other statistics
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 goodnessof-t hypothesis H : F = F for some known continuous df F against the alternative of location. Well-known distribution-free statistics for this hypothesis are the Kolmogorov statistic Dn and omega-square statistic ω n. e following table presents their local absolute eciency 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 happens 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 eciencies 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 NeymanPearson approach. In contrast with Bahadur eciency, 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 exponential 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 eciency is that most two-sided tests like Kolmogorov and Cramér-von Mises tests are asymptotically optimal, and hence this kind of eciency cannot discriminate between them. On the other hand, under some regularity conditions the onesided tests like linear rank tests can be compared on the basis of their indices, and their Hodges–Lehmann e- ciency coincides locally with Bahadur eciency, see details in Nikitin (). Coupled with three “basic” approaches to the ARE calculation described above, intermediate approaches are also possible if the transition to the limit occurs simultaneously for two parameters at a controlled way.us emerged the Cherno ARE introduced by Cherno (), 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 eciency of tests was applied in recent years to more general problems. 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 stationary 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. Professor Nikitin is the author of the text Asymptotic eciency of nonparametric tests, Cambridge University Press, NY, , and has authored more than papers, in many international journals, in the eld of Asymptotic eciency of statistical tests, large deviations of test statistics and nonparametric Statistics. Cross References 7Asymptotic Relative Eciency in Estimation 7Cherno-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 :–
Asymptotic,Higher Order Borovkov A,Mogulskii A (1993)Large deviations and testing of truly approximating a target quantity up to the claimed A statistical hypotheses.Siberian Adv Math 2(3,4);3(1,2) degree of accuracy. Chernoff H(1952)A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations.Ann Math Stat Classical asymptotics is based on the notion of asymp- 23:493-507 totic distribution,often derived from the central limit Dembo A,ZeitouniO(1998)Large deviations techniques and appli- theorem (see Central Limit Theorems),and usually the cations,2nd edn.Springer,New York approximations are correct up to O(n-2),where n is Deuschel J-D.Stroock D(1989)Large deviations.Academic.Boston the sample size.Higher order asymptotics provides refine- Groeneboom P.Oosterhoff J (1981)Bahadur efficiency and small ments based on asymptotic expansions of the distribution sample efficiency.Int Stat Rev 49:127-141 Hodges J,Lehmann EL(1956)The efficiency of some nonparametric or density function of an estimator of a parameter.They are competitors of the f-test.Ann Math Stat 26:324-335 rooted in the Edgeworth theory,which is itselfa refinement Kakizawa Y(2005)Bahadur exact slopes of some tests for spectral of the central limit theorem.The theory of higher order densities.J Nonparametric Stat 17:745-764 asymptotic is very much related with the corresponding Kallenberg WCM(1983)Intermediate efficiency,theory and exam to Approximations to distributions treated as well in this ples.Ann Stat 11:170-182 Kallenberg WCM(1982)Chernoff efficiency and deficiency.Ann Encyclopedia. Stat10:583-594 When higher order asymptotic is correct up to Nikitin Y (1995)Asymptotic efficiency of nonparametric tests. o(n),it is second order asymptotic.When further Cambridge University Press,Cambridge terms are picked up,so that the asymptotic is correct up Otsu T(2010)On Bahadur efficiency of empirical likelihood.JEcon to o(n),it is third order asymptotic.In his pioneer- 157:248-256 Pitman EJG(1949)Lecture notes on nonparametric statistical infer- ing papers,C.R.Rao coined the term second order effi- ence.Columbia University,Mimeographed ciency for a concept that would now is called third order Puhalskii A,Spokoiny V(1998)On large-deviation efficiency in efficiency.The new terminology is essentially owing to statistical inference.Bernoulli 4:203-272 Pfanzagl and Takeuchi. Serfling R(1980)Approximation theorems of mathematical statis- tics.Wiley,New York Serfling R(2010)Asymptotic relative efficiency in estimation.In: About the Author Lovric M (ed)International encyclopedia of statistical sciences. Professor Abril is co-editor of the Revista de la Sociedad Springer Argentina de Estadistica (Journal of the Argentinean Sta- Taniguchi M(2001)On large deviation asymptotics of some tests in tistical Society). time series.J Stat Plann Inf 97:191-200 Van der Vaart AW (1998)Asymptotic statistics.Cambridge Univer- sity Press,Cambridge Cross References Wieand HS(1976)A condition under which the Pitman and Bahadur Approximations to Distributions approaches to efficiency coincide.Ann Statist 4:1003-1011 Edgeworth Expansion References and Further Reading Abril JC(1985)Asymptotic expansions for time series problems Asymptotic,Higher Order with applications to moving average models.PhD thesis.The London School of Economics and Political Science,University of London,England JUAN CARLOS ABRIL Barndorff-Nielsen O,Cox DR(1979)Edgeworth and saddle-point President of the Argentinean Statistical Society,Professor approximations with statistical applications.J R Stat Soc B Universidad Nacional de Tucuman and Consejo Nacional 41:279-312 de Investigaciones Cientificas y Tecnicas,San Miguel de Daniels HE(1954)Saddlepoint approximations in statistics.Ann Math Stat 25:631-650 Tucuman,Argentina Durbin J(1980)Approximations for the densities of sufficient esti- mates.Biometrika 67:311-333 Higher order asymptotic deals with two sorts of closely Feller W(1971)An introduction to probability theory and its appli- related things.First,there are questions of approxima- cations,vol 2,2nd edn.Wiley,New York Ghosh JK(1994)Higher order Asymptotic.NSF-CBMS Regional tion.One is concerned with expansions or inequalities Conference Series in Probability and Statistics,4.Hayward and for a distribution function.Second,there are inferential Alexandria:Institute of Mathematical Statistics and American issues.These involve,among other things,the applica- Statistical Association tion of the ideas of the study of higher order efficiency, Pfanzagl J(1979)Asymptotic expansions in parametric statistical admissibility and minimaxity.In the matter of expansions, theory.In:Krishnaiah PR(ed)Developments in statistics,vol.3. Academic,New York,pp 1-97 it is as important to have usable,explicit formulas as a rig- Rao CR (1961)Asymptotic efficiency and limiting information.In orous proof that the expansions are valid in the sense of Proceedings of Fourth Berkeley Symposium on Mathematical
Asymptotic, Higher Order A A Borovkov A, Mogulskii A () Large deviations and testing of statistical hypotheses. Siberian Adv Math (, ); (, ) Chernoff H () A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations. Ann Math Stat :– Dembo A, Zeitouni O () Large deviations techniques and applications, nd edn. Springer, New York Deuschel J-D, Stroock D () Large deviations. Academic, Boston Groeneboom P, Oosterhoff J () Bahadur efficiency and small sample efficiency. Int Stat Rev :– Hodges J, Lehmann EL () The efficiency of some nonparametric competitors of the t-test. Ann Math Stat :– Kakizawa Y () Bahadur exact slopes of some tests for spectral densities. J Nonparametric Stat :– Kallenberg WCM () Intermediate efficiency, theory and examples. Ann Stat :– Kallenberg WCM () Chernoff efficiency and deficiency. Ann Stat :– Nikitin Y () Asymptotic efficiency of nonparametric tests. Cambridge University Press, Cambridge Otsu T () On Bahadur efficiency of empirical likelihood. J Econ :– Pitman EJG () Lecture notes on nonparametric statistical inference. Columbia University, Mimeographed Puhalskii A, Spokoiny V () On large-deviation efficiency in statistical inference. Bernoulli :– Serfling R () Approximation theorems of mathematical statistics. Wiley, New York Serfling R () Asymptotic relative efficiency in estimation. In: Lovric M (ed) International encyclopedia of statistical sciences. Springer Taniguchi M () On large deviation asymptotics of some tests in time series. J Stat Plann Inf :– Van der Vaart AW () Asymptotic statistics. Cambridge University Press, Cambridge Wieand HS () A condition under which the Pitman and Bahadur approaches to efficiency coincide. Ann Statist :– Asymptotic, Higher Order Juan Carlos Abril President of the Argentinean Statistical Society, Professor Universidad Nacional de Tucumán and Consejo Nacional de Investigaciones Cientícas y Técnicas, San Miguel de Tucumán, Argentina Higher order asymptotic deals with two sorts of closely related things. First, there are questions of approximation. One is concerned with expansions or inequalities for a distribution function. Second, there are inferential issues. ese involve, among other things, the application of the ideas of the study of higher order eciency, admissibility and minimaxity. In the matter of expansions, it is as important to have usable, explicit formulas as a rigorous proof that the expansions are valid in the sense of truly approximating a target quantity up to the claimed degree of accuracy. Classical asymptotics is based on the notion of asymptotic distribution, oen derived from the central limit theorem (see 7Central Limit eorems), and usually the approximations are correct up to O(n −/ ), where n is the sample size. Higher order asymptotics provides renements based on asymptotic expansions of the distribution or density function of an estimator of a parameter.ey are rooted in the Edgeworth theory, which is itself a renement of the central limit theorem. e theory of higher order asymptotic is very much related with the corresponding to Approximations to distributions treated as well in this Encyclopedia. When higher order asymptotic is correct up to o(n −/ ), it is second order asymptotic. When further terms are picked up, so that the asymptotic is correct up to o(n − ), it is third order asymptotic. In his pioneering papers, C. R. Rao coined the term second order e- ciency for a concept that would now is called third order eciency. e new terminology is essentially owing to Pfanzagl and Takeuchi. About the Author Professor Abril is co-editor of the Revista de la Sociedad Argentina de Estadística (Journal of the Argentinean Statistical Society). Cross References 7Approximations to Distributions 7Edgeworth Expansion References and Further Reading Abril JC () Asymptotic expansions for time series problems with applications to moving average models. PhD thesis. The London School of Economics and Political Science, University of London, England Barndorff-Nielsen O, Cox DR () Edgeworth and saddle-point approximations with statistical applications. J R Stat Soc B :– Daniels HE () Saddlepoint approximations in statistics. Ann Math Stat :– Durbin J () Approximations for the densities of sufficient estimates. Biometrika :– Feller W () An introduction to probability theory and its applications, vol , nd edn. Wiley, New York Ghosh JK () Higher order Asymptotic. NSF-CBMS Regional Conference Series in Probability and Statistics, . Hayward and Alexandria: Institute of Mathematical Statistics and American Statistical Association Pfanzagl J () Asymptotic expansions in parametric statistical theory. In: Krishnaiah PR (ed) Developments in statistics, vol. . Academic, New York, pp – Rao CR () Asymptotic efficiency and limiting information. In Proceedings of Fourth Berkeley Symposium on Mathematical