On the coefficient of multiple determination in a linear regression model

Summary

Consider ap-variate random vectorX=(X ₁,…,X _p ),p≧2, with an unknown meanμ ∈ R ^p and an unknown dispersion matrix σ(p.d.). ForX ₍₂₎ =(X ₂,…,X _p )′, the regression ofX ₁ onX ₍₂₎ is defined asE(X ₁ |X ₍₂₎. The multiple correlation coefficient betweenX ₁ andX ₍₂₎, denoted by ρ_1·23...p , is the simple correlation coefficient betweenX ₁ and its best linear fitX _1·23...p =β ₁+β ₂ X ₂+...β _p X _p byX ₍₂₎ whereβ _i ’s are regression coefficients. The parameter λ=ρ ² _1·23...p is called the coefficient of multiple determination and it indicates the extent of the true contribution of the explanatory variablesX ₂…X _p in explaining the response variableX ₁ 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 (N _p (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.