Comments on: Model-free model-fitting and predictive distributions

Abstract

I enjoyed reading this paper, which presents a new look to the problem of developing predictive distributions under very general conditions. The key idea of this new method is transforming the data to iid observations exploiting a semiparametric model (such as a mean-scale one) or the probability integral transformation (PIT) when a completely nonparametric model is preferred under a smoothness class assumption. In this last sense the term model-free prediction is coined by the author. This approach of transformation to iid (pseudo) residuals is parallel to the usual modeling strategy conducted in parametric statistics to isolate the random component in the data from the information contained in the conditioning variables. Then, resampling can be used naturally to construct valid confidence intervals using the empirical distribution of these residuals which are iid by construction. To accommodate (possibly infinite dimensional) parameter estimation effects, one-leave-out transformations are proposed by the author to replicate properly in finite samples the associated uncertainty to predicting a new design point. This principle provides its own point predictions inverting the transformation and using an appropriate solution for a given loss function, so it is quite different from trying to build confident intervals for a given prediction mechanism, for which specific techniques or adjustments could be necessary. Though in simulations it appears that such predictions behave similarly to the usual nonparametric estimators, it could be interesting to further investigate in detail the properties of these new point estimates (of conditional expectations), both asymptotically and in finite samples.