Central limit theorems for the $L_{p}$-error of smooth isotonic estimators

Abstract

We investigate the asymptotic behavior of the Lp-distance between a monotone function on a compact interval and a smooth estimator of this function. Our main result is a central limit theorem for the Lp-error of smooth isotonic estimators obtained by smoothing a Grenander-type estimator or isotonizing the ordinary kernel estimator. As a preliminary result we establish a similar result for ordinary kernel estimators. Our results are obtained in a general setting, which includes estimation of a monotone density, regression function and hazard rate. We also perform a simulation study for testing monotonicity on the basis of the L2-distance between the kernel estimator and the smoothed Grenander-type estimator.MSC 2010 subject classifications: Primary 62G20; secondary 62G10.

On the Lp error of the Grenander‐type estimator in the Cox model

“…However, that is outside the scope of this article and will be addressed in the future. The current theoretical results provide also the basis for studying the L p -error of smoothed monotone estimators, which can be used to test monotonicity as well (see Lopuhaä & Musta, 2019).…”