Optimal prediction for quantiles and probabilities

A prediction statement about a future random variable should always be accompanied by a measure of uncertainty and in particular it should be based on a so-called predictive distribution. The choice of a predictive distribution should be driven by the aim of the prediction itself, that usually is to compute predictive quantiles or predictive probabilities. Instead, it is common in the practice to use the same predictive distribution for both targets. The goal of this paper is twofold. First we provide a critical review of some important properties that a predictive distribution should possess, encouraging the correct choice of a predictive distribution in accordance to the predictive goal. Secondly, we propose suitable predictive distribution functions that satisfy the considered properties, to a high order of approximation. These are all in the form of asymptotic corrections of the plug-in distribution function. Some of them are already known in literature and some others are new proposals. In particular we introduce a new predictive distribution that may be useful to obtain calibrated predictions for the probabilities of a future random variable. The computation of the associated asymptotic expansion, when too difficult, can be substituted by simpler bootstrap procedures. The different predictive distribution functions are also compared in terms of mean squared error. As a side result, we show that the predictive distribution that minimises the mean squared error is also optimal with respect to the continuous ranked probability score. Some examples illustrate how to compute the new calibrated predictive distribution. Moreover, several simulation studies clearly show how the performance of different predictive distributions depends on the aim of the prediction and that the improvement can be substantial, in particular for the case of small and medium sample sizes.