ESTIMATION OF THE COEFFICIENT OF MULTIPLE DETERMINATION

NABENDU PAL^{1} AND WOOI K. LIM^{2}

1Department of Mathematics, University of Southwestern Louisiana, Lafayette, LA 70504, U.S.A. 2Department of Mathematics, University of New Orleans, New Orleans, LA 70148, U.S.A.

(Received March 17, 1997; revised August 7, 1997)

Abstract. Assume that we have iid observations on the random vector X = (X_{1}, …, X_{p})′ following a multivariate normal distribution N_{p}(μ, Σ) where both μ ∈ R^{p} and Σ(p.d.) are unknown. Let ρ_{1·23···p} denote the multiple correlation coefficient between X_{1} and (X_{2}, …, X_{p})′. The parameter λ = ρ_{1·23···p}^{2}, called the multiple coefficient of determination, indicates the proportion of variability in X_{1} explained by its best linear fit based on (X_{2}, …, X_{p})′. In this paper we consider the point estimation of λ under the ordinary squared error loss function. The usual estimators (MLE, UMVUE) have complicated risk expressions and hence it is quite difficult to get exact decision-theoretic results. We therefore follow the asymptotic decision theoretic approach (as done by Ghosh and Sinha (1981, Ann. Statist., 9, 1334–1338)) and study ‘Second Order Admissibility’ of various estimators including the usual ones.

Key words and phrases: Multiple correlation coefficient, loss function, risk function, second order admissibility.