(Received March 17, 1997; revised August 7, 1997)
Abstract. Assume that we have iid observations on the random vector X = (X1, ..., Xp)' following a multivariate normal distribution Np(mu, Sigma) where both mu \in \calR p and Sigma(p.d.) are unknown. Let rho1 · 23 ··· p denote the multiple correlation coefficient between X11 and (X2, ..., Xp)'. The parameter lambda = rho21 · 23 ··· p, called the multiple coefficient of determination, indicates the proportion of variability in X1 explained by its best linear fit based on (X2, ..., Xp)'. In this paper we consider the point estimation of lambda 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.
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