## The Annals of Statistics

### Locally adaptive regression splines

#### 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

#### 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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