## The Annals of Statistics

### Nonparametric regression, confidence regions and regularization

#### Abstract

In this paper we offer a unified approach to the problem of nonparametric regression on the unit interval. It is based on a universal, honest and nonasymptotic confidence region $\mathcal{A}_{n}$ which is defined by a set of linear inequalities involving the values of the functions at the design points. Interest will typically center on certain simplest functions in $\mathcal{A}_{n}$ where simplicity can be defined in terms of shape (number of local extremes, intervals of convexity/concavity) or smoothness (bounds on derivatives) or a combination of both. Once some form of regularization has been decided upon the confidence region can be used to provide honest nonasymptotic confidence bounds which are less informative but conceptually much simpler.

#### Article information

Source
Ann. Statist., Volume 37, Number 5B (2009), 2597-2625.

Dates
First available in Project Euclid: 17 July 2009

https://projecteuclid.org/euclid.aos/1247836662

Digital Object Identifier
doi:10.1214/07-AOS575

Mathematical Reviews number (MathSciNet)
MR2541440

Zentralblatt MATH identifier
1173.62023

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G15: Tolerance and confidence regions 62G20: Asymptotic properties

#### Citation

Davies, P. L.; Kovac, A.; Meise, M. Nonparametric regression, confidence regions and regularization. Ann. Statist. 37 (2009), no. 5B, 2597--2625. doi:10.1214/07-AOS575. https://projecteuclid.org/euclid.aos/1247836662

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