Conditional predictive inference for high-dimensional stable algorithms
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We investigate generically applicable and intuitively appealing prediction intervals based on leave-one-out residuals. The conditional coverage probability of the proposed interval, given the observations in the training sample, is close to the nominal level, provided that the underlying algorithm used for computing point predictions is sufficiently stable under the omission of single feature-response pairs. Our results are based on a finite sample analysis of the empirical distribution function of the leaveone- out residuals and hold in a non-parametric setting with only minimal assumptions on the error distribution. To illustrate our results, we also apply them to high-dimensional linear predictors, where we obtain uniform asymptotic conditional validity as both sample size and dimension tend to infinity at the same rate. These results show that despite the serious problems of resampling procedures for inference on the unknown parameters (cf. Bickel and Freedman, 1983; El Karoui and Purdom, 2015; Mammen, 1996), leave-one-out methods can be successfully applied to obtain reliable predictive inference even in high dimensions.
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