On the weighted multivariate Wilcoxon rank regression estimate

Zhou (2010) introduced a multivariate Wilcoxon regression estimate which possesses some nice properties: computational ease, asymptotic normality and high efficiency. However, it is sensitive to the leverage points. To circumvent this problem, we propose a weighted multivariate Wilcoxon regression estimate. Under some regularity conditions, the asymptotic normality is established. We further study the robustness of the proposed estimate through the influence function. By properly choosing the weight functions, our results show that the corresponding estimate can have bounded influence function on both response and covariates.     

Wilcoxon signed rank test using ranked-set sample

Ranked-set sampling is useful when measurements are destructive or costly to obtain but ranking of the observations is relatively easy. The Wilcoxon signed rank test statistic based on the ranked-set sample is considered. We compared the asymptotic relative efficiencies of the RSS Wilcoxon signed rank test statistic with respect to the SRS Wilcoxon signed rank test statistic and the RSS sign test statistic. Throughout the ARE’s, the proposed test statistic is superior to the SRS Wilcoxon signed rank test statistic and the RSS sign test statistic.     

Competitors of the Wilcoxon signed rank test

Summary Distribution-free statistics are proposed for one-sample location test, and are compared with the Wilcoxon signed rank test. It is shown that one of the statistics is superior to the Wilcoxon test in terms of approximate Bahadur efficiency. And we compare that statistic with the Wilcoxon test from the viewpoint of asymptotic expansion of power function under contiguous alternatives.     

Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors

We develop optimal rank-based procedures for testing affine-invariant linear hypotheses on the parameters of a multivariate general linear model with elliptical VARMA errors. We propose a class of optimal procedures that are based either on residual (pseudo-)Mahalanobis signs and ranks, or on absolute interdirections and lift-interdirection ranks, i.e., on hyperplane-based signs and ranks. The Mahalanobis versions of these procedures are strictly affine-invariant, while the hyperplane-based ones are asymptotically affine-invariant. Both versions generalize the univariate signed rank procedures proposed by Hallin and Puri (J. Multivar. Anal. 50 (1994) 175), and are locally asymptotically most stringent under correctly specified radial densities. Their AREs with respect to Gaussian procedures are shown to be convex linear combinations of the AREs obtained in Hallin and Paindaveine (Ann. Statist. 30 (2002) 1103; Bernoulli 8 (2002) 787) for the pure location and purely serial models, respectively. The resulting test statistics are provided under closed form for several important particular cases, including multivariate Durbin-Watson tests, VARMA order identification tests, etc. The key technical result is a multivariate asymptotic linearity result proved in Hallin and Paindaveine (Asymptotic linearity of serial and nonserial multivariate signed rank statistics, submitted).     

The multivariate signed Bollobás-Riordan polynomial

We generalise the signed Bollobás-Riordan polynomial of S. Chmutov and I. Pak [S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial, Mos. Math. J. 7(3) (2007), 409-418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial, J. Combin. Theory, Ser. B 99(3) (2009), 617-638. arXiv:0711.3490, doi:10.1016/j.jctb.2008.09.007] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.     

Semi-aligned rank tests

Summary The distribution-free test based on semi-aligned rankings for no treatment effects in a two-way layout, with unequal number of replications in each cell is considered. The asymptotic χ-square distribution of the test statistic under the null hypothesis is derived. The Pitman asymptotic relative efficiency of the test (i) based on semi-aligned rankings with respect to the test (ii) based on within-block rankings, is shown to be larger than one as the number of blocks tends to infinity. Also the asymptotic properties of linear rank statistics (i) and (ii) are investigated and the asymptotic relative efficiency of the test (i) with respect to the test (ii) is again shown to be larger than one.     

Asymptotically distribution-free aligned rank order tests for composite hypotheses for general multivariate linear models

For general multivariate linear models, a composite hypothesis does not usually induce invariance of the joint distribution under appropriate groups of transformations, so that genuinely distribution-free tests do not usually exist. For this purpose, some aligned rank order statistics are incorporated in the proposal and study of a class of asymptotically distribution-free tests. Tests for the parallelism of several multiple regression surfaces are also considered. Finally the optimal properties of these tests are discussed.Work supported by the Air Force Office of Scientific Research, AFSC, USAF, Contract Nos: AFOSR-74-2736 and AFOSR-76-2927. Reproduction in whole or part is permitted for any purpose of the U.S. Government     

On the rate of convergence in the central limit theorem for signed rank statistics

The problem of the rate of convergence of the cumulative distribution function of the one-sample rank order statistics S_N⁺ to the limiting normal one is studied for the case when the underlying observations are independent but not necessarily identically distributed. The results obtained are then used to derive a strong law and functional form of a central limit theorem for S_N⁺     

Two-sample inference based on one-sample Wilcoxon signed rank statistics

A two-sample test is studied which rejects the null hypothesis of equal population medians when two Wilcoxon distribution free confidence intervals are disjoint. A confidence interval for the difference in population medians is constructed by subtracting the endpoints of two one-sample confidence intervals. Two different ways to select the one-sample intervals are presented. A solution that specifies equal confidence coefficients for the one-sample intervals is recommended. All solutions are shown to have the same asymptotic (Pitman) efficiency as the Mann-Whitney two-sample test.     

On the admissibility of stable spherical multivariate tests

This paper deals with correlation tests from the class of spherical tests introduced by Läuter (Biometrics 52 (1996) 964). These methods provide an alternative to classical MANOVA approaches and are particularly useful in small samples. Following a brief introduction of the spherical tests, it is shown that the so-called principal component correlation test is admissible in this class. A Bayesian approach is used to prove this result.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.