The Annals of Statistics

Interaction Spline Models and Their Convergence Rates

Zehua Chen

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Abstract

We consider interaction splines which model a multivariate regression function $f$ as a constant plus the sum of functions of one variable (main effects), plus the sum of functions of two variables (two-factor interactions), and so on. The estimation of $f$ by the penalized least squares method and the asymptotic properties of the models are studied in this article. It is shown that, under some regularity conditions on the data points, the expected squared error averaged over the data points converges to zero at a rate of $O(N^{-2m/(2m + 1)})$ as the sample size $N \rightarrow \infty$ if the smoothing parameters are appropriately chosen, where $m$ is a measure of the assumed smoothness of $f.$

Article information

Source
Ann. Statist., Volume 19, Number 4 (1991), 1855-1868.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348374

Digital Object Identifier
doi:10.1214/aos/1176348374

Mathematical Reviews number (MathSciNet)
MR1135152

Zentralblatt MATH identifier
0738.62065

JSTOR
links.jstor.org

Subjects
Primary: 62H12: Estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Keywords
Prediction mean squared error reproducing kernel Hilbert space kernel matrix

Citation

Chen, Zehua. Interaction Spline Models and Their Convergence Rates. Ann. Statist. 19 (1991), no. 4, 1855--1868. doi:10.1214/aos/1176348374. https://projecteuclid.org/euclid.aos/1176348374


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