Prediction intervals with controlled length in the heteroscedastic Gaussian regression

We tackle the problem of building a prediction interval in heteroscedastic Gaussian regression. We focus on prediction intervals with constrained expected length in order to guarantee interpretability of the output. In this framework, we derive a closed form expression of the optimal prediction interval that allows for the development a data-driven prediction interval based on plug-in. The construction of the proposed algorithm is based on two samples, one labeled and another unlabeled. Under mild conditions, we show that our procedure is asymptotically as good as the optimal prediction interval both in terms of expected length and error rate. In particular, the control of the expected length is distribution-free. We also derive rates of convergence under smoothness and the Tsybakov noise conditions. We conduct a numerical analysis that exhibits the good performance of our method. It also indicates that even with a few amount of unlabeled data, our method is very effective in enforcing the length constraint.