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ESTIMATION OF THE COEFFICIENT OF

MULTIPLE DETERMINATION

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

^{1} *Department of Mathematics, University of Southwestern Louisiana,*

Lafayette, LA 70504, U.S.A.

^{2} *Department 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}(*mu*, *Sigma*) where both
*mu* \in \cal**R**^{ p} and *Sigma*(*p.d.*) are unknown.
Let
*rho*_{1 · 23 ··· p} denote the multiple correlation
coefficient between *X*1_{1} and (*X*_{2}, ..., *X*_{p})^{'}.
The parameter
*lambda* = *rho*^{2}_{1 · 23 ··· p,}
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 *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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