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

In a recent paper Dette, Neumeyer and Pilz (H. Dette, N. Neumeyer, and K.F. Pilz, A simple nonparametric estimator of a strictly monotone regression function, Bernoulli 12 (2006), pp. 469–490) proposed a new nonparametric estimate of a smooth 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 types of smooth monotone estimates. In this note, we provide a more refined asymptotic analysis of the monotone regression estimate. In the case where the regression function is increasing but only once continuously differentiable, we prove the asymptotic normality of an appropriately standardised 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 small. Therefore, the rate of convergence of the new estimate is worse than the rate n−1/3 of the (unsmoothed) monotone least squares estimate. On the other hand, if the derivative of the regression function is Hölder continuous, the rate of convergence of the new estimate is faster than n−1/3.