A simple nonparametric estimator of a strictly monotone regression function

Abstract

A new method for monotone estimation of a regression function is proposed, which is potentially attractive to users of conventional smoothing methods. The main idea of the new approach is to construct a density estimate from the estimated values $\hat{m}(i/N)$ ($i=1,\dots ,N$) of the regression function and to use these `data' for the calculation of an estimate of the inverse of the regression function. The final estimate is then obtained by a numerical inversion. Compared to the currently available techniques for monotone estimation the new method does not require constrained optimization. We prove asymptotic normality of the new estimate and compare the asymptotic properties with the unconstrained estimate. In particular, it is shown that for kernel estimates or local polynomials the bandwidths in the procedure can be chosen such that the monotone estimate is first-order asymptotically equivalent to the unconstrained estimate. We also illustrate the performance of the new procedure by means of a simulation study.