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THE CONVERGENCE RATES OF EMPIRICAL BAYES

ESTIMATION IN A MULTIPLE LINEAR REGRESSION MODEL

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LAISHENG WEI^{ 1} AND SHUNPU ZHANG^{ 2}

^{1} *Department of Mathematics, University of Science and
Technology of China,*

Hefei, Anhui, 230026, China

^{2} *Department of Mathematics, Hangzhou Normal College,
Zhejiang,
310036, China*
(Received January 17, 1994; revised July 27, 1994)

**Abstract.**
Empirical Bayes (EB) estimation of the
parameter vector *theta*=(*beta*', *sigma*^{2})' in a multiple linear
regression model *Y* = X*beta*+*epsilon* is considered, where *beta* is
the vector of regression coefficient, *epsilon* ~ *N*(0,*sigma*^{2}*I*) and
*sigma*^{2} is unknown. In this paper, we have constructed the EB
estimators of *theta* by using the kernel estimation of multivariate
density function and its partial derivatives. Under suitable
conditions it is shown that the convergence rates of the EB
estimators are *O*(*n*^{-(lambda k-1)(k-2)/k(2k+p+1)}), where the
natural number *k* __>__ 3, 1/3 < *lambda* < 1, and *p* is the
dimension of vector *beta*.

*Key words and phrases*:
Empirical Bayes estimation,
multiple linear regression model, convergence rates.

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