  We investigate the problem of regression where one is allowed to abstain from
predicting. We refer to this framework as regression with reject option as an
extension of classification with reject option. In this context, we focus on
the case where the rejection rate is fixed and derive the optimal rule which
relies on thresholding the conditional variance function. We provide a
semi-supervised estimation procedure of the optimal rule involving two
datasets: a first labeled dataset is used to estimate both regression function
and conditional variance function while a second unlabeled dataset is exploited
to calibrate the desired rejection rate. The resulting predictor with reject
option is shown to be almost as good as the optimal predictor with reject
option both in terms of risk and rejection rate. We additionally apply our
methodology with kNN algorithm and establish rates of convergence for the
resulting kNN predictor under mild conditions. Finally, a numerical study is
performed to illustrate the benefit of using the proposed procedure.
