Statistical procedures based on signed ranks ink samples with unequal variances

Abstract

Ink samples with unequal variances, test procedures based on signed ranks for the homogeneity ofk location parameters are proposed. The asymptotic χ²-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 one-sample signed rank tests relative to thet-test. A simulation study for the goodness of the χ²-approximate of significance points is done. Then, surprisingly it can be seen that the χ²-approximate for the critical points of the proposed tests is better than that of Kruskal-Wallis test and the Welch-type test. NextR-estimators and weighted least squares estimators for common mean ofk samples under the homogeneity ofk location parameters are compared in the same way as the test case. Furthermore, positive-part shrinkage versions ofR-estimators for thek location parameters are considered along with a modified James-Stein estimation rule. The asymptotic distributional risks of the usualR-estimators, the positive-part shrinkageR-estimators (PSRE's), and the preliminary test and shrinkageR-versions under an arbitrary quadratic loss are derived. Under Mahalanobis loss, it is shown that the PSRE's dominate the otherR-estimators fork≥4. A simulation study leads strong support to the claims that the PSRE's dominate the other typeR-estimators and they are robust about outliers.