The Annals of Statistics

Locally adaptive regression splines

Enno Mammen and Sara van de Geer

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Abstract

Least squares penalized regression estimates with total variation penalties are considered. It is shown that these estimators are least squares splines with locally data adaptive placed knot points. The definition of these variable knot splines as minimizers of global functionals can be used to study their asymptotic properties. In particular, these results imply that the estimates adapt well to spatially inhomogeneous smoothness. We show rates of convergence in bounded variation function classes and discuss pointwise limiting distributions. An iterative algorithm based on stepwise addition and deletion of knot points is proposed and its consistency proved.

Article information

Source
Ann. Statist., Volume 25, Number 1 (1997), 387-413.

Dates
First available in Project Euclid: 10 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034276635

Digital Object Identifier
doi:10.1214/aos/1034276635

Mathematical Reviews number (MathSciNet)
MR1429931

Zentralblatt MATH identifier
0871.62040

Subjects
Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions

Keywords
Nonparametric curve estimation penalized least squares splines local adaptivity rates of convergence

Citation

Mammen, Enno; van de Geer, Sara. Locally adaptive regression splines. Ann. Statist. 25 (1997), no. 1, 387--413. doi:10.1214/aos/1034276635. https://projecteuclid.org/euclid.aos/1034276635


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