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Abstract:
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Artemiou and Li (to appear: Statistica Sinica) gave a probabilistic explanation of a natural phenomenon that is frequently observed but whose reason is not well understood. That is, in a regression setting, the response (Y) is often highly correlated with the leading principal components of the predictor (X) even though there seems no logical reason for this connection. In this work we make three extensions of the previous result. First, we extend the result in a nonlinear regression setting, where the link function is unknown. Second, in both the linear and nonlinear setting we extend the result to multivariate response Y and for fixed covariances. Finally, in the linear case, we establish a general inequality using the information I=E(log (f(X,Y) / f(X)f(Y))).
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