Local Bahadur efficiency of rank tests for the independence problem

In previous papers [Approximate and local Bahadur efficiency of linear rank tests in the two-sample problem, Ann. Statist.7, 1246–1255, 1979; Local comparison of linear rank tests in the Bahadur sense, Metrika, 1979] the author developed for linear rank tests of the one-sample symmetry and the k-sample problem (k ≥ 2) a theory of local comparison, based on the concept of Bahadur efficiency. In the present article this theory is carried over to rank tests of the independence problem.     

On some tests of homogeneity of variances

Summary In their paper [1], Ansari and Bradley discussed a two-sample rank test for dispersions and suggested the desirability of extending their results to the problem of several samples. In this paper, besides generalizing their results, we provide a few additional non-parametric tests, which include, among others, the multi-sample analogues of the two-sample normal scores test of dispersion and the tests considered by Mood [6], Klotz [4], and Siegel and Tukey [10]. The asymptotic distributions of the proposed test statistics are derived by an application of the author’s theorem [7]. The asymptotic efficiencies of these tests relative to one another and theF test ([9], pp. 83–87) are computed in the standard fashion along the lines of the author’s paper [7]. The work represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under Sloan Foundation Grant for statistics and under U.S. Navy Contract Nonr-285 (38).     

Practical tests for randomized complete block designs

A class of affine-invariant test statistics, including a sign test and a related family of signed-rank tests, is proposed for randomized complete block designs with one observation per treatment. This class is obtained by using the transformation-retransformation approach of Chakraborty, Chaudhuri and Oja along with a directional transformation due to Tyler. Under the minimal assumption of directional symmetry of the underlying distribution, the null asymptotic distribution of the sign test statistic is shown to be chi-square with p-1 degrees of freedom. The same null distribution is also proved for the family of signed-rank statistics under the assumption of symmetry of the underlying distribution. The Pitman asymptotic relative efficiencies of the tests, relative to Hotelling-Hsu's T² are established. Several score functions are discussed including a simple linear score function and the optimal normal score function. The test based on the linear score function is compared to the other members of this family and other statistics in the literature through efficiency calculations and Monte Carlo simulations. This statistic has an excellent performance over a wide range of distributions and for small as well as large dimensions.     

Symmetrized approximate score rank tests for the two-sample case

Rank test statistics for the two-sample problem are based on the sum of the rank scores from either sample. However, a critical difference can occur when approximate scores are used since the sum of the rank scores from sample 1 is not equal to minus the sum of the rank scores from sample 2. By centering and scaling as described in Hajek and Sidak (1967, Theory of Rank Tests, Academic Press, New York) for the uncensored data case the statistics computed from each sample become identical. However such symmetrized approximate scores rank statistics have not been proposed in the censored data case. We propose a statistic that treats the two approximate scores rank statistics in a symmetric manner. Under equal censoring distributions the symmetric rank tests are efficient when the score function corresponds to the underlying model distribution. For unequal censoring distributions we derive a useable expression for the asymptotic variance of our symmetric rank statistics.     

A local breakdown property of robust tests in linear regression

The breakdown slope, as a useful summary measure of local stability for estimators and test statistics, has been studied recently by He, Simpson, and Protnoy (1990, J. Amer. Statist. Assoc., 85). It is shown here that all regression estimates based on residuals alone in linear models have zero breakdown slopes in contamination neighborhoods, even though they can have breakdown points as high as one-half. The breakdown functions of tests based on the S-estimation are investigated. It is also shown that the Generalized M-estimators can have better local breakdown robustness. One way to obtain regression estimators with desirable local and global breakdown properties is discussed.     

A Bahadur efficiency comparison between one and two sample rank statistics and their sequential rank statistic analogues

One and two sample rank statistics are shown in general to be more efficient in the Bahadur sense than their sequential rank statistic analogues as defined by Mason (1981, Ann. Statist.9 424–436) and Lombard (1981, South African Statist. J.15 129–152), even though the two families of statistics (those based on full ranks and those based on sequential ranks) have the same Pitman efficiency against local alternatives. In the process, general results on large deviation probabilities and laws of large numbers for statistics based on sequential ranks are obtained.     

Asymptotic expansions for two-stage rank tests

Stein's two-stage procedure produces a t-test which can realize a prescribed power against a given alternative, regardless of the unknown variance of the underlying normal distribution. This is achieved by determining the size of a second sample on the basis of a variance estimate derived from the first sample. In the paper we introduce a nonparametric competitor of this classical procedure by replacing the t-test by a rank test. For rank tests, the most precise information available are asymptotic expansions for their power to order n ^-1, where n is the sample size. Using results on combinations of rank tests for sub-samples, we obtain the same level of precision for the two-stage case. In this way we can determine the size of the additional sample to the natural order and moreover compare the nonparametric and the classical procedure in terms of expected additional numbers of observations required.     

A formula for deficiency (L-andR-tests)

In this paper an analogue of the formulas [D. M. Chibisov,Teor. Veroyatn. Primen.,30, 269–288 (1985);Izv. Akad. Nauk UzSSR,6, 23–30 (1982)] for the difference between the power of a given asymptotically efficient test and that of the most powerful test is justified for one-sample L-and R-tests, i.e., tests based on linear combinations of order statistics and linear rank statistics. This formula directly yields the Hodges-Lehmann deficiency of corresponding tests. A general theorem is stated which is applied to L-and R-tests. The explicit expressions given by this formula for L- and R-tests are also presented. The expression related to R-tests agrees with the one obtained in [W. Albers, P. J. Bickel, and W. R. Van Zwet,Ann. Statist.,4, 108–156 (1976);6, 1170–1171 (1978)]. We present here a nontechnical (heuristic) proof of these results. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-1446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

Local powers of two-sample and multi-sample rank tests for Lehmann's contaminated alternative

Summary Local powers of two- andk-sample rank tests under alternatives of contaminated distributions are investigated. It is shown that the rank tests based on normal scores and Wilcoxon scores are superior to thet-test or theF-test for many choices of alternatives of contaminated distributions and that the values of the asymptotic relative efficiency of the rank test based on Wilcoxon scores with respect to the normal scores are about one in all the investigated cases.     

On the limiting Pitman efficiency of some rank tests of independence

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 [17], Ann. Statist.4 1003–1011) on coincidence of the limiting Pitman efficiency and the local (as the alternative approaches the hypothesis) approximate Bahadur efficiency. [7], 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.     

Two tests for multivariate normality based on the characteristic function

We present two tests for multivariate normality. The presented tests are based on the Lévy characterization of the normal distribution and on the BHEP tests. The tests are affine invariant and consistent. We obtain the asymptotic null distribution of the test statistics using some results about generalized one-sample U-statistics, which are of independent interest.      

Nonparametric tests for multiple regression under progressive censoring

For continuous observations from time-sequential studies, suitable Cramér-von Mises and Kolmogorov-Smirnov types of (nonparametric) statistics (based on linear rank statistics) for testing hypotheses on some multiple-regression models are proposed and studied. The asymptotic theory of these tests is provided for both the null and (local) alternative hypotheses situations and is based on the weak convergence of suitable rank order processes (on the D[0, 1] space) to certain functions of Brownian motions. Bahadur efficiency results are also presented. Empirical values of the percentile points of the null distributions of the proposed test statistics, obtained through simulation studies, are also provided.     

Tests for normality based on density estimators of convolutions

Recent results show that densities of convolutions can be estimated by local U-statistics at the root-n rate in various norms. Motivated by this and the fact that convolutions of normal densities are normal, we introduce new tests for normality which use as test statistics weighted L₁-distances between the standard normal density and local U-statistics based on standardized observations. We show that such test statistics converge at the root-n rate and determine their limit distributions as functionals of Gaussian processes. We also address a choice of bandwidth. Simulations show that our tests are competitive with other tests of normality.     

Quasi-duality for the rings of generalized power series *

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered commutative monoid which is also artinian. For any bimodule _AM_B , we construct a bimodule _A[[S]]M[S]_B[[S]] and prove that _AM_B defines a quasi-duality if and only if the bimodule _A[[S]]M[S]_B[[S]] defines a quasi-duality. As a corollary, it is shown that if a ring A has a quasi-duality then the ring A[[S]] of generalized power series over A has a quasi-duality.     

The limiting normality of the test statistic for the two-sample problem induced by a convex sum distance

Critchlow (1992, J. Statist. Plann. Inference, 32, 325–346) proposed a method of a unified construction of a class of rank tests. In this paper, we introduce a convex sum distance and prove the limiting normality of the test statistics for the two-sample problem derived by his method.     

On multivariate signed rank tests

Summary As a test for the hypothesis of equality of means in a sample ofn matched pairs of observations from a single bivariate population in(x, y), Wilcoxon ([4], [5]) proposed a signed rank sum testU involving the differencesz=x\t-y. Tables of valuesUs have been provided forn of 50 or less and four significance levelsε. The following paper considers some generalizations of the Wilcoxon test for the case of matched bivariate or trivariate populations. These tests utilize certain properties of thep-variate normal distribution when integrated over certain regions for the cases:p=2,3.     

Tests d'égalité contre l'alternative de croisement de deux fonctions de survie

We propose two tests to test the hypothesis of the equality of two survival functions against the alternative of crossing of these functions for right-censured data. The asymptotic laws of test statistics are given. The powers of the tests are calculated. To cite this article: V. Bagdonavi?ius et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Likelihood Ratio Tests for Principal Components

A particular class of tests for the principal components of a scatter matrix Σ is proposed. In the simplest case one wants to test whether a given vector is an eigenvector of Σ corresponding to its largest eigenvalue. The test statistics are likelihood ratio statistics for the classical Wishart model, and critical values are obtained parametrically as well as nonparametrically without making any assumptions on the eigenvalues of Σ. Still, the tests have asymptotic properties similar to those of classical procedures and are asymptotically admissible and optimal in some sense.     

Generalized canonical analysis for time series

Canonical correlation analysis is shown to be equivalent to the problem of estimating a linear regression matrix, B₀, of less than full rank. After reparameterizing B₀ some estimates of the new parameters, obtained by solving an eigenvalue problem and closely related to canonical correlations and vectors, are found to be consistent and efficient when the residuals are mutually independent. When the residuals are generated by an autocorrelated, stationary time series these estimates are still consistent and obey a central limit theorem, but they are no longer efficient. Alternative, more general estimates are suggested which are efficient in the presence of serial correlation. Asymptotic theory and iterative computational procedures for these estimates are given. A likelihoodratio test for the rank of B₀ is seen to be an extension of a familiar test for canonical correlations.