Abstract
We investigate the asymptotic behavior of the $L_{p}$-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 $L_{p}$-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 $L_{2}$-distance between the kernel estimator and the smoothed Grenander-type estimator.
Highlights
The property of monotonicity plays an important role when dealing with survival data or regression relationships
In this paper we investigate the Lp-error of smooth isotonic estimators obtained by kernel smoothing the Grenander-type estimator or by isotonizing the ordinary kernel estimator
Since the behavior of these estimators is closely related to the behavior of ordinary kernel estimators, we first establish a central limit theorem for the Lp-error of ordinary kernel estimators for a monotone function on a compact interval
Summary
The property of monotonicity plays an important role when dealing with survival data or regression relationships. Our main results are central limit theorems for the Lp-error of smooth isotonic estimators for a monotone function on a compact interval. Since the behavior of these estimators is closely related to the behavior of ordinary kernel estimators, we first establish a central limit theorem for the Lp-error of ordinary kernel estimators for a monotone function on a compact interval This extends the work by [10] on the Lp-error of densities that are smooth on the whole real line, but is of interest by itself. Since the isotonization step is performed last, the estimator is inconsistent at the boundaries For this reason, we can only obtain a central limit theorem for the Lp-error on a sub-interval that approaches the whole support, as n diverges to infinity.
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