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Several new methods have been proposed for performing valid inference aftermodel selection. An older method is sampling splitting: use part of the datafor model selection and part for inference. In this paper we revisit samplesplitting combined with the bootstrap (or the Normal approximation). We showthat this leads to a simple, assumption-free approach to inference and weestablish results on the accuracy of the method. In fact, we find new bounds onthe accuracy of the bootstrap and the Normal approximation for generalnonlinear parameters with increasing dimension which we then use to assess theaccuracy of regression inference. We show that an alternative, called the imagebootstrap, has higher coverage accuracy at the cost of more computation. Wedefine new parameters that measure variable importance and that can be inferredwith greater accuracy than the usual regression coefficients. There is ainference-prediction tradeoff: splitting increases the accuracy and robustnessof inference but can decrease the accuracy of the predictions.

Bootstrapping and Sample Splitting For High-Dimensional, Assumption-Free Inference