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We show that under different moment bounds on the underlying variables, bootstrap approximation to the large deviation probabilities of standardized sample sum, based on independent random variables, is valid for a wider zone of n, the sample size, compared to the classical normal tail probability approximation. As an application, different notions of efficiency for statistical tests are considered from Bayesian point of view. In particular, efficiency due to Pitman (1938) [11], Chernoff (1952) [1], and Bayes risk efficiency due to Rubin and Sethuraman (1965) [12] turn out to be special cases with the choice of the weight function; i.e., prior density times loss.

Classical univariate measures of asymmetry such as Pearson’s (mean-median)/σ or (mean-mode)/σ often measure the standardized distance between two separate location parameters and have been widely used in assessing univariate normality. Similarly, measures of univariate kurtosis are often just ratios of two scale measures. The classical standardized fourth moment and the ratio of the mean deviation to the standard deviation serve as examples. In this paper we consider tests of multinormality which are based on the Mahalanobis distance between two multivariate location vector estimates or on the (matrix) distance between two scatter matrix estimates, respectively. Asymptotic theory is developed to provide approximate null distributions as well as to consider asymptotic efficiencies. Limiting Pitman efficiencies for contiguous sequences of contaminated normal distributions are calculated and the efficiencies are compared to those of the classical tests by Mardia. Simulations are used to compare finite sample efficiencies. The theory is also illustrated by an example. peerReviewed

In this paper multivariate extensions of the Friedman and Page tests for the comparison of several treatments are introduced. Related unadjusted and adjusted treatment effect estimates for the multivariate response variable are also found and their properties discussed. The test statistics and estimates are analogous to the traditional univariate methods. In test constructions, the univariate ranks are replaced by multivariate spatial ranks (J. Nonparam. Statist. 5 (1995) 201). Asymptotic theory is developed to provide approximations for the limiting distributions of the test statistics and estimates. Limiting efficiencies of the tests and treatment effect estimates are found in the multivariate normal and t distribution cases. The tests are rotation invariant only, but affine invariant versions can be easily constructed. The theory is illustrated by an example.

The shortcoming of a test is the difference between the maximal attainable power and the power of the test under consideration. Vanishing shortcoming, when the number of observations tends to infinity, is therefore an optimality property of a test. Other familiar optimality criteria are based on the asymptotic relative efficiency of the test. The relations between these optimality criteria are investigated. It turns out that vanishing shortcoming is seemingly slightly stronger than first-order efficiency, but in regular cases there is equivalence. The results are in particular applied on tests for goodness-of-fit.

The generalized likelihood ratio (GLR) test proposed by Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] and Fan and Yao [Nonlinear Time Series: Nonparametric and Parametric Methods (2003) Springer] is a generally applicable nonparametric inference procedure. In this paper, we show that although it inherits many advantages of the parametric maximum likelihood ratio (LR) test, the GLR test does not have the optimal power property. We propose a generally applicable test based on loss functions, which measure discrepancies between the null and nonparametric alternative models and are more relevant to decision-making under uncertainty. The new test is asymptotically more powerful than the GLR test in terms of Pitman's efficiency criterion. This efficiency gain holds no matter what smoothing parameter and kernel function are used and even when the true likelihood function is available for the GLR test., Published in at http://dx.doi.org/10.1214/13-AOS1099 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

New test statistics are proposed for testing whether two random vectors are independent. Gieser and Randles, as well as Taskinen, Kankainen, and Oja have introduced and discussed multivariate extensions of the quadrant test of Blomqvist. This article serves as a sequel to this work and presents new multivariate extensions of Kendall's tau and Spearman's rho statistics. Two different approaches are discussed. First, interdirection proportions are used to estimate the cosines of angles between centered observation vectors and between differences of observation vectors. Second, covariances between affine-equivariant multivariate signs and ranks are used. The test statistics arising from these two approaches appear to be asymptotically equivalent if each vector is elliptically symmetric. The spatial sign versions are easy to compute for data in common dimensions, and they provide practical, robust alternatives to normal-theory methods. Asymptotic theory is developed to approximate the finite-sample null distributions as well, as to calculate limiting Pitman efficiencies. Small-sample null permutation distributions are also described. A simple simulation study is used to compare the proposed tests with the classical Wilks test. Finally, the theory is illustrated by an example. peerReviewed

A new affine invariant extension of the quadrant test statistic Blomqvist (Ann. Math. Statist. 21 (1950) 593) based on spatial signs is proposed for testing the hypothesis of independence. In the elliptic case, the new test statistic is asymptotically equivalent to the interdirection test by Gieser and Randles (J. Amer. Statist. Assoc. 92 (1997) 561) but is easier to compute in practice. Limiting Pitman efficiencies and simulations are used to compare the test to the classical Wilks’ test. peerReviewed

Design measures maximizing local power of asymptotically uniformly most powerful (AUMP) tests about the value of logit $P$ outside the observation space are characterized.

usseldorf It is shown that the global power function of any nonparametric test is flat on balls of alternatives except for alternatives coming from a finite dimensional subspace. The present benchmark is here the upper one-sided (or two-sided) envelope power function. Every choice of a test fixes a priori a finite dimensional region with high power. It turns out that also the level points are far away fromthe corresponding Neym an–Pearson test level points except for a finite number of orthogonal directions of alternatives. For certain submodels the result is independent of the underlying sample size. In the last section the statistical consequences and special goodness of fit tests are discussed. 1. Introduction. Omnibus tests are commonly used if the specific structure of certain nonparametric alternatives is unknown. Among other justifications, it turns out that they typically are consistent against fixed alternatives and √ n-consistent under sequences of local alternatives of sample size n .F or these reasons, people often trust in goodness of fit tests and these are frequently applied to data of finite sample size. On the other hand, every asymptotic approximation should be understood as an approximation of the underlying finite sample case. Thus the statistician likes to distinguish and to compare the power of different competing tests. The present paper offers a concept for the comparison and justification of different tests by their power functions and level points. It is shown that under certain circumstances every test has a preference for a finite dimensional space of alternatives. Apart fromthis space, the power function is almost flat on balls of alternatives. There exists no test which pays equal attention to an infinite number of orthogonal alternatives. The results do not only hold for asymptotic models but they also hold for concrete alternatives on the real line at finite sample size and their level points uniformly for the sample size. The results are not surprising. Every statistician knows that it is impossible to separate an infinite sequence of different parameters simultaneously if only a finite number of observations is available. The conclusions of the results are two-fold. 1. The statistician should analyze the goodness of fit tests of his computer package in order to get some knowledge and an impression about their preferences.

This paper investigates a new family of statistics based on Burbea–Rao divergence for testing goodness-of-fit. Under the simple and composite null hypotheses the asymptotic distribution of these tests is shown to be chi-squared. For composite hypothesis, the unspecified parameters are estimated by maximum likelihood as well as minimum Burbea–Rao divergence

We consider general chi-square goodness-of-fit test statistics for randomly censored data--call these generalized Pearson statistics--which are nonnegative definite quadratic forms in the cell frequencies obtained from the product-limit estimator, allowing random cells and general estimators of nuisance parameters. This class of statistics generalizes the class studied by Moore and Spruill in the no censoring case. The large sample behavior of these statistics under the null hypothesis and local alternatives is presented. The chi-square type statistics based on the observed cell frequencies obtained from the product-limit estimator are members of this class for which the quadratic form is selected to produce a chi-square asymptotic null distribution. The generalized Pearson statistic and the statistic by Akritas for a simple hypothesis are compared on the basis of asymptotic relative Pitman efficiency. It is shown that neither statistic dominates the other. The efficiencies are shown to depend on the degree of censoring and the number of cells. For heavily censored data, the Akritas statistic is superior to the generalized Pearson statistic. In the uncensored case, the Akritas statistic, which does not reduce to the Pearson statistic, is not as good as the Pearson statistic in the sense of Pitman efficiency.

Let X 1 ,…, X n be a sequence of independent and identically distributed random variables with an unknown underlying continuous cumulative distribution function F . Relative to this unknown distribution function suppose one would like to test a null hypothesis concerning the goodness of fit of F to some distribution function using symmetric functions of sample spacings. In some applications the null hypothesis is simple while in others it may be composite. In this article we present the large sample theory of tests based on symmetric functions of sample spacings under composite null hypotheses and contiguous alternatives. It is shown that these test statistics have the same asymptotic distribution in the case when parameters must be estimated from the sample as in the case when parameters are specified. Optimal goodness of fit tests are also constructed for these hypotheses.

The limiting (as the significance level approaches 0) Pitman efficiency of a new “regression-based” rank test of independence to Kendall's tau and Spearman's rho is derived. The result is based on a version of Wieand's theorem (1976 , Ann. Statist. 4 1003–1011) on coincidence of the limiting Pitman efficiency and the local (as the alternative approaches the hypothesis) approximate Bahadur efficiency. Kiefer and Wolfowitz (1958 , Trans. Amer. Math. Soc. 87 , 173–186) result is applied to verify the main assumption of Wieand's paper. This approach is shown to be useful in some other situations, also.

In an earlier paper, the present author ( Sarkar (1979) , Calcutta Statist. Assoc. Bull. 28 , 47–56) proposed a similar test for a mean testing problem with additional observations on a set of correlated auxiliary variables. This idea has been extended here to cover some multivariate linear regression testing problems with the same type of additional observations on a set of correlated auxiliary variables.

Since their introduction by Dwass (1957) and Barnard (1963), Monte Carlo tests have attracted considerable attention. The aim of this paper is to give a unified approach that covers the case of an arbitrary null distribution in order to study the statistical properties of Monte Carlo tests under the null hypothesis and under the alternative. For finite samples we obtain bounds for the power of the Monte Carlo test with the original test that allow determination of the required simulation effort. Furthermore the concept of asymptotic (resp. local asymptotic) relative Pitman efficiency (ARPE, resp. LARPE) is adapted to Monte Carlo tests for the study of their asymptotic behaviour. The normal limit case is investigated in more detail, leading to explicit formulas for ARPE and LARPE.

Sequential tests for one-sided hypotheses are compared, asymptotically, with non-sequential counterparts. An analog of Pitman efficiency is obtained, as is another comparison that has no purely non-sequential analog. With these methods of comparison, the limiting relative efficiency of the sequential test is never less than one and for most parameter values, it is infinite. An asymptotic notion of minimal relative efficiency is also considered.

The approximate Bahadur efficiency and the Pitman efficiency for hypothesis testing problems are considered. A theorem is stated and proved which gives a condition under which the existence of the limiting (as the alternative approaches the hypothesis) approximate Bahadur efficiency implies the existence of the limiting (as the significance level approaches 0) Pitman efficiency and the equality of the two limits. Several examples are then given to show how the theorem may be used in computing previously unknown limiting Pitman efficiencies using the Bahadur approach.

The classical problem of choice of number of classes in testing goodness of fit is considered for a class of alternatives, for the chi-square and likelihood ratio statistics. Pitman and Bahadur efficiencies are used to compare the two statistics and also to analyse the effect for each statistic of changing the number of classes for the case where the number of classes increases asymptotically with the number of observations. Overall, the results suggest that if the class of alternatives is suitably restricted the number of classes should not be very large.

A general approach to Pitman efficiency as a limit of the ratio of sample sizes is presented. The results can be used especially to derive the Pitman efficiency of tests based on asymptotically $\chi^2$-distributed statistics with different degrees of freedom.

General results on the limiting equivalence of local and nonlocal measures of efficiency are obtained. Why equivalence occurs in so many testing and estimation problems is clarified. Uniformity of the convergence is a key point. The concepts of Frechet- and Hadamard-type differentiability, which imply uniformity, play an important role. The theory is applied to tests based on linear rank statistics, showing equivalence of the local limit of exact Bahadur efficiency and Pitman efficiency. As a second application, the relation between the inaccuracy rate and the asymptotic variance of $L$-estimators is investigated.

Various selection problems based on sample means have been considered by Bechhofer (Ann. Math. Statist. (1954), 16-39). Parallel selection procedures based on one-sample Hodges-Lehmann estimators (Ann. Math. Statist. (1963), 598-611) are proposed in this paper. These procedures possess more desirable finite sample properties and equivalent asymptotic properties as compared to rival procedures considered by Lehmann (Math. Ann. (1963), 268-275) and Puri and Puri (Ann. Math. Statist. (1969), 619-632).