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In statistical prediction, classical approaches for model selection and modelevaluation based on covariance penalties are still widely used. Most of theliterature on this topic is based on what we call the "Fixed-X" assumption,where covariate values are assumed to be nonrandom. By contrast, it is oftenmore reasonable to take a "Random-X" view, where the covariate values areindependently drawn for both training and prediction. To study theapplicability of covariance penalties in this setting, we propose adecomposition of Random-X prediction error in which the randomness in thecovariates contributes to both the bias and variance components. Thisdecomposition is general, but we concentrate on the fundamental case of leastsquares regression. We prove that in this setting the move from Fixed-X toRandom-X prediction results in an increase in both bias and variance. When thecovariates are normally distributed and the linear model is unbiased, all termsin this decomposition are explicitly computable, which yields an extension ofMallows' Cp that we call $RCp$. $RCp$ also holds asymptotically for certainclasses of nonnormal covariates. When the noise variance is unknown, pluggingin the usual unbiased estimate leads to an approach that we call $\hat{RCp}$,which is closely related to Sp (Tukey 1967), and GCV (Craven and Wahba 1978).For excess bias, we propose an estimate based on the "shortcut-formula" forordinary cross-validation (OCV), resulting in an approach we call $RCp^+$.Theoretical arguments and numerical simulations suggest that $RCP^+$ istypically superior to OCV, though the difference is small. We further examinethe Random-X error of other popular estimators. The surprising result we getfor ridge regression is that, in the heavily-regularized regime, Random-Xvariance is smaller than Fixed-X variance, which can lead to smaller overallRandom-X error.

From Fixed-X to Random-X Regression: Bias-Variance Decompositions, Covariance Penalties, and Prediction Error Estimation