Cross-validation is a resampling procedure that provides a point estimate of generalization error for any predictive model. Cross-validation is widely used for model selection and evaluation. Uncertainty in the cross-validation estimate is challenging to quantify, and estimation of its variance is known to require multiple runs of the entire resampling procedure. Nested cross-validation computes a prediction interval for the generalization error for a given model and training dataset by resampling the entire cross-validation procedure but incurs extraordinary computational cost. We provide an efficient method for computing the nested cross-validation prediction interval using only a single model fit for some penalized regression models including ridge regression, spline smoothing, and some functional regression models. We characterize when our proposed method should be expected to out-perform resampling-based nested cross-validation in various scaling regimes as well as in finite samples. Experiments for functional principle components regression demonstrate non-trivial cases in which our proposed method improves run times substantially.

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