THE CONVERGENCE RATES OF EMPIRICAL BAYES ESTIMATION IN A MULTIPLE LINEAR REGRESSION MODEL

THE CONVERGENCE RATES OF EMPIRICAL BAYES
ESTIMATION IN A MULTIPLE LINEAR REGRESSION MODEL

LAISHENG WEI¹ AND SHUNPU ZHANG²

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', sigma²)' in a multiple linear regression model Y = Xbeta+epsilon is considered, where beta is the vector of regression coefficient, epsilon ~ N(0,sigma²I) and sigma² 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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