STATISTICAL PROCEDURES BASED ON SIGNED RANKS IN k SAMPLES WITH UNEQUAL VARIANCES

TAKA-AKI SHIRAISHI

Department of Mathematics, Yokohama City University, Yokohama 236, Japan

(Received November 2, 1991; revised April 16, 1992)

Abstract. In k samples with unequal variances, test procedures based on signed ranks for the homogeneity of k location parameters are proposed. The asymptotic χ^{2}-distribution of the test statistics is shown. It is found that the asymptotic relative efficiency of the rank tests relative to Welch’s test (1951, Biometrika, 38, 330–336) under local alternatives agrees with that of the onesample signed rank tests relative to the t-test. A simulation study for the goodness of the χ^{2}-approximate of significance points is done. Then, surprisingly it can be seen that the χ^{2}-approximate for the critical points of the proposed tests is better than that of Kruskal-Wallis test and the Welch-type test. Next R-estimators and weighted least squares estimators for common mean of k samples under the homogeneity of k location parameters are compared in the same way as the test case. Furthermore, positive-part shrinkage versions of Restimators for the k location parameters are considered along with a modified James-Stein estimation rule. The asymptotic distributional risks of the usual R-estimators, the positive-part shrinkage R-estimators (PSRE’s), and the preliminary test and shrinkage R-versions under an arbitrary quadratic loss are derived. Under Mahalanobis loss, it is shown that the PSRE’s dominate the other R-estimators for k ≥ 4. A simulation study leads strong support to the claims that the PSRE’s dominate the other type R-estimators and they are robust about outliers.

Key words and phrases: Hypothesis-testing, parameter estimation, Behrens-Fisher’s problem, asymptotic relative efficiency, asymptotic distributional risk, modified James-Stein rule, simulation study.