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ROBUST ESTIMATION OF COMMON REGRESSION

COEFFICIENTS UNDER SPHERICAL SYMMETRY

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T. KUBOKAWA^{1}, C. ROBERT^{2} AND A. K. Md. E. SALEH^{3}

^{1} *Department of Mathematical Engineering and Information Physics, Faculty of Engineering,*

University of Tokyo, Bunkyo-ku, Tokyo 113, Japan

^{2} *I.S.U.P.-L.S.T.A., Universite de Paris VI, Tour 45, 4, place Jussieu,*

75252 Paris Cedex 05, France

^{3} *Department of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6*
(Received October 20, 1989; revised October 17, 1990)

**Abstract.**
Consider the problem of estimating the common
regression coefficients of two linear regression models where the
two distributions of the errors may be different and unknown.
Under the spherical symmetry assumption, the paper proves the
superiority of a Graybill-Deal type combined estimator and the
further improvement by the Stein effect which were exhibited by
Shinozaki (1978, *Comm. Statist. Theory Methods*, **7**,
1421-1432) in the normal case. This shows the robustness of
the dominations since the conditions for the dominations are
independent of the errors distributions.

*Key words and phrases*:
Elliptically contoured distribution,
heteroscedastic linear model, Stein problem, common mean,
Graybill-Deal estimator.

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