  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.
