The Annals of Statistics

Nonparametric Regression Under Qualitative Smoothness Assumptions

Enno Mammen

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Abstract

We propose a new nonparametric regression estimate. In contrast to the traditional approach of considering regression functions whose $m$th derivatives lie in a ball in the $L_\infty$ or $L_2$ norm, we consider the class of functions whose $(m - 1)$st derivative consists of at most $k$ monotone pieces. For many applications this class seems more natural than the classical ones. The least squares estimator of this class is studied. It is shown that the speed of convergence is as fast as in the classical case.

Article information

Source
Ann. Statist., Volume 19, Number 2 (1991), 741-759.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348118

Digital Object Identifier
doi:10.1214/aos/1176348118

Mathematical Reviews number (MathSciNet)
MR1105842

Zentralblatt MATH identifier
0737.62039

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62J02: General nonlinear regression 62E20: Asymptotic distribution theory

Keywords
Nonparametric regression simple qualitative curve characteristics isotonic and concave regression estimation of the shape of a function

Citation

Mammen, Enno. Nonparametric Regression Under Qualitative Smoothness Assumptions. Ann. Statist. 19 (1991), no. 2, 741--759. doi:10.1214/aos/1176348118. https://projecteuclid.org/euclid.aos/1176348118


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