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

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

1Department of Mathematics, University of Science and Technology of China,

Hefei, Anhui, 230026, China

2Department 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 θ = (β′, σ^{2})′ in a multiple linear regression model Y = Xβ + ϵ is considered, where β is the vector of regression coefficient, ϵ ∼ N(0, σ^{2}I) and σ^{2} is unknown. In this paper, we have constructed the EB estimators of θ 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^{−(λk−1)(k−2)/k(2k+p+1)}), where the natural number k ≥ 3, \frac{1}{3} < λ < 1, and p is the dimension of vector β.

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