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STATISTICAL PROCEDURES BASED ON SIGNED RANKS

IN *k* SAMPLES WITH UNEQUAL VARIANCES

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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 *chi*^{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
one-sample signed rank tests relative to the *t*-test. A simulation study
for the goodness of the *chi*^{2}-approximate of significance points is
done. Then, surprisingly it can be seen that the *chi*^{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
*R*-estimators 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.

**Source**
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