Monotone nonparametric regression with random design

Abstract

In this paper we study the nonparametric least squares estimator of a regression function in a random design setting under the constraint that this function is monotone, say, nonincreasing. The errors are not assumed conditionally i.i.d. given the observation points. In particular, this includes the case of conditional heteroscedasticity and the case of the current status model. The $$ \mathbb{L}_p $$ -error is shown to be of order n −p/3 and asymptotically Gaussian with explicit asymptotic mean and variance.