Estimation of the Coefficient of Multiple Determination

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$_{1 \cdot 23 \cdots p}$ denote the multiple correlation coefficient between X$_1$ and (X$_2$,...,X$_p$)′. The parameter λ = ρ$_{1 \cdot 23 \cdots p}^2$, 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.