## The Annals of Statistics

### Asymptotics for $M$-Type Smoothing Splines

Dennis D. Cox

#### Abstract

Limit theorems giving rates of convergence of nonparametric regression estimates obtained from smoothing splines are proved. The main emphasis is on nonlinear, robust smoothing splines, but new results are obtained for the usual (linear) case. It is assumed that the knots become asymptotically uniform in a vague sense. Convergence of derivatives is also investigated. The main mathematical tools are a linearization of the robust smoothing spline, and an approximation of the linear smoothing spline utilizing the Green's function of an associated boundary value problem.

#### Article information

Source
Ann. Statist., Volume 11, Number 2 (1983), 530-551.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346159

Digital Object Identifier
doi:10.1214/aos/1176346159

Mathematical Reviews number (MathSciNet)
MR696065

Zentralblatt MATH identifier
0519.62034

JSTOR
Cox, Dennis D. Asymptotics for $M$-Type Smoothing Splines. Ann. Statist. 11 (1983), no. 2, 530--551. doi:10.1214/aos/1176346159. https://projecteuclid.org/euclid.aos/1176346159