On the coefficient of multiple determination in a linear regression model

Consider ap-variate random vectorX=(X1,…,Xp),p≧2, with an unknown meanμ ∈ Rp and an unknown dispersion matrix σ(p.d.). ForX(2) =(X2,…,Xp)′, the regression ofX1 onX(2) is defined asE(X1|X(2). The multiple correlation coefficient betweenX1 andX(2), denoted by ρ1·23...p, is the simple correlation coefficient betweenX1 and its best linear fitX1·23...p=β1+β2X2+...βpXp byX(2) whereβi’s are regression coefficients. The parameter λ=ρ21·23...p is called the coefficient of multiple determination and it indicates the extent of the true contribution of the explanatory variablesX2…Xp in explaining the response variableX1 through a linear regression model. In this article we address the problem of efficient estimation of λ under the Pitman Nearness Criterion (PNC) as well as the Stochastic Domination Criterion (SDC) assuming thatX follows a multivariate normal distribution (Np(μ, σ)). We have proposed several competing shrinkage estimators which seem to outperform the usual maximum likelihood estimator by wide margins. Finally, simulation results and two real life data sets (from environmental studies) have been used to demonstrate the effectiveness of our proposed estimators.