## Electronic Journal of Statistics

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

#### 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.

#### Article information

Source
Electron. J. Statist., Volume 13, Number 1 (2019), 1031-1098.

Dates
Received: July 2018
First available in Project Euclid: 5 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1554429624

Digital Object Identifier
doi:10.1214/19-EJS1550

Mathematical Reviews number (MathSciNet)
MR3935844

Zentralblatt MATH identifier
07056146

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62G10: Hypothesis testing

#### Citation

Lopuhaä, Hendrik P.; Musta, Eni. Central limit theorems for the $L_{p}$-error of smooth isotonic estimators. Electron. J. Statist. 13 (2019), no. 1, 1031--1098. doi:10.1214/19-EJS1550. https://projecteuclid.org/euclid.ejs/1554429624

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