A note on estimating a monotone regression by combining kernel and density estimates

Abstract

In a recent paper Dette, Neumeyer and Pilz (2005) proposed a new nonparametric estimate of a monotone regression function. This method is based on a non-decreasing rearrangement of an arbitrary unconstrained nonparametric estimator. Under the assumption of a twice continuously differentiable regression function the estimate is first order asymptotic equivalent to the unconstrained estimate and other type of monotone estimates. In this note we provide a more refined asymptotic analysis of the monotone regression estimate. It is shown that in the case of a non-decreasing regression function the new method produces an estimate with nearly the same Lp-norm as the given function for any p ≥ 1. Moreover, in the case, where the regression function is increasing but only once continuously differentiable we prove asymptotic normality of an appropriately standardized version of the estimate, where the asymptotic variance is of order n^{−2/3−ε}, the bias is of order n^{−1/3+ε} and ε > 0 is arbitrarily small. Therefore the rate of convergence of the new estimate is arbitrarily close to the rate of the estimate obtained from monotone least squares estimation, but the asymptotic distribution of the new estimate is substantially simpler.