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 e›ciency is the classical notion used most oŸen 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 continu￾ous distribution and under the alternative dišer 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 sam￾ples 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 innitely better than the parametric test of Student but their ARE never falls below .. See analogous results in Seržing () 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 e›ciency is /π  ≈ .. In numerical comparisons, the Pitman e›ciency appear to be more relevant for moderate sample sizes than other e›ciencies Groeneboom and Oosterhoš (). On the other hand, Pitman ARE can be insu›cient for the comparison of tests. Suppose, for instance, that we have a normally distributed sample with the mean θ and vari￾ance  and we are testing H : θ =  against A : θ > . Let compare two signicance 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 e›ciency, these two tests are equivalent. On the contrary, Bahadur e›ciency e B t,X¯ (θ) is strictly less than  for any θ > . If the condition of asymptotic normality fails, consid￾erable di›culties 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 e›ciency 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 observa￾tions 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 dened 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 nec￾essary 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 com￾pared in the literature with the 7Cramér–Rao inequality in the estimation theory. erefore the absolute (nonrelative) Bahadur e›ciency of the sequence {Tn} can be dened 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 ). OŸen 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 e›ciency. e indisputable merit of Bahadur e›ciency 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