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

Abstract

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). 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) procedures are also calculated under the contaminated normal model.

Keywords:

• Mann–Whitney–Wilcoxon
• Robust estimation
• Shift parameter
• Smoothing
• Two-sample problem

Mathematics Subject Classification:

• Primary 62G05
• Secondary 62G35

Acknowledgment

The authors wish to thank Prof. N. Balakrishnan and the referees for their helpful comments and suggestions.