Smoothed Mann–Whitney–Wilcoxon Procedure for Two-Sample Location Problem

This study is mainly concerned with estimating a shift parameter in the two-sample location problem. The proposed Smoothed Mann–Whitney–Wilcoxon method smooths the empirical distribution functions of each sample by using convolution technique, and it replaces unknown distribution functions F(x) and G(x ? Δ₀) with the new smoothed distribution functions F _s (x) and G _s (x ? Δ₀), respectively. The unknown shift parameter Δ₀ is estimated by solving the gradient function S _n (Δ) with respect to an arbitrary variable Δ. The asymptotic properties of the new estimator are established under some conditions that are similar to the Generalized Wilcoxon procedure proposed by Anderson and Hettmansperger (1996 Anderson , G. F. , Hettmansperger , T. P. ( 1996 ). Generalized Wilcoxon methods for the one and two-sample location models . In: Brunner , E. , Denker , M. , eds. Research Developments in Probability and Statistics: Festschrift in Honor of Madan L. Puri on the Occasion of his 65th Birthday . Zeist, The Netherlands : VSP BV , pp. 303 – 317 . [Google Scholar]). Some of these properties are asymptotic normality, asymptotic level confidence interval, and hypothesis testing for Δ₀. Asymptotic relative efficiency of the proposed method with respect to the least squares, Generalized Wilcoxon and Hodges and Lehmann (1963 Hodges , J. L. , Lehmann , E. L. ( 1963 ). Estimates of location based on rank tests . Ann. Mathemat. Statist. 34 : 598 – 611 .[Crossref] , [Google Scholar]) procedures are also calculated under the contaminated normal model.     

Robust procedures for multi-sample location problems with unequal group variances

Robust procedures are proposed for testing the equality of several group means without assuming the equality of group variances. These statistics are obtained by modifying Welch's W and Brown-Forsythe's F^* using a trimmed mean and a sine-wave M estimator.Approximate distributions of these new statistics are obtained under normality. Their performances are evaluated by Monte Carlo sampling experiments under various long-tailed symmetric distributions     

Modified Wilcoxon–Mann–Whitney Test and Power Against Strong Null

The Wilcoxon–Mann–Whitney (WMW) test is a popular rank-based two-sample testing procedure for the strong null hypothesis that the two samples come from the same distribution. A modified WMW test, the Fligner–Policello (FP) test, has been proposed for comparing the medians of two populations. A fact that may be under-appreciated among some practitioners is that the FP test can also be used to test the strong null like the WMW. In this article, we compare the power of the WMW and FP tests for testing the strong null. Our results show that neither test is uniformly better than the other and that there can be substantial differences in power between the two choices. We propose a new, modified WMW test that combines the WMW and FP tests. Monte Carlo studies show that the combined test has good power compared to either the WMW and FP test. We provide a fast implementation of the proposed test in an open-source software. Supplementary materials for this article are available online.     

An exact density-based empirical likelihood ratio test for paired data

The Wilcoxon rank-sum test and its variants are historically well-known to be very powerful nonparametric decision rules for testing no location difference between two groups given paired data versus a shift alternative. In this title, we propose a new alternative empirical likelihood (EL) ratio approach for testing the equality of marginal distributions given that sampling is from a continuous bivariate population. We show that in various shift alternative scenarios the proposed exact test is superior to the classic nonparametric procedures, which may break down completely or are frequently inferior to the density-based EL ratio test. This is particularly true in the cases where there is a nonconstant shift under the alternative or the data distributions are skewed. An extensive Monte Carlo study shows that the proposed test has excellent operating characteristics. We apply the density-based EL ratio test to analyze real data from two medical studies.