A comparative study of monotone nonparametric kernel estimates

Abstract

In this paper we present a detailed numerical comparison of three monotone nonparametric kernel regression estimates, which isotonize a nonparametric curve estimator. The first estimate is the classical smoothed isotone estimate of Brunk [Brunk, H.D., 1955, Maximum likelihood estimates of monotone parameters. The Annals of Mathematical Statistics, 26, 607–616.]. The second method has recently been proposed by Hall and Huang [Hall, P. and Huang, L.-S., 2001, Nonparametric kernel regression subject to monotonicity constraints. The Annals of Statistics, 29, 624–647.] and modifies the weights of a commonly used kernel estimate such that the resulting estimate is monotone. The third estimate was recently proposed by Dette et al. [Dette, H., Neumeyer, N. and Pilz, K.F., 2003, A simple non-parametric estimator of a monotone regression function. Technical report, Department of Mathematics. Available online at: http://www.ruhr-uni-bochum.de/mathematik3/preprint.htm] and combines density and regression estimation techniques to obtain a monotone curve estimate of the inverse of the isotone regression function. The three concepts are briefly reviewed and their finite sample properties are studied by means of a simulation study. Although all estimates are first-order asymptotically equivalent (provided that the unknown regression function is isotone) some differences for moderate sample sizes are observed.