The Annals of Statistics

Asymptotic Integrated Mean Square Error Using Least Squares and Bias Minimizing Splines

Girdhar G. Agarwal and W. J. Studden

Full-text: Open access

Abstract

Let $S^d_k$ be the set of $d$th order splines on $\lbrack 0, 1 \rbrack$ having $k$ knots $\xi_1 < \xi_2 \cdots < \xi_k$. We consider the estimation of a sufficiently smooth response function $g$, using $n$ uncorrelated observations, by an element $s$ of $S^d_k$. For large $n$ and $k$ we have discussed the asymptotic behavior of the integrated mean square error (IMSE) for two types of estimators: (i) the least squares estimator and (ii) a bias minimizing estimator. The asymptotic expression for IMSE is minimized with respect to three variables. (i) the allocation of observation (ii) the displacement of knots $\xi_1 < \cdots < \xi_k$ and (iii) number of knots.

Article information

Source
Ann. Statist. Volume 8, Number 6 (1980), 1307-1325.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176345203

Digital Object Identifier
doi:10.1214/aos/1176345203

Mathematical Reviews number (MathSciNet)
MR594647

Zentralblatt MATH identifier
0522.62032

JSTOR
links.jstor.org

Subjects
Primary: 62K05: Optimal designs
Secondary: 62J05: Linear regression 62F10: Point estimation 41A50: Best approximation, Chebyshev systems 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]

Keywords
Optimal design least square estimator bias minimizing estimator approximation $B$-splines Bernoulli polynomial $L_2$-projection operator

Citation

Agarwal, Girdhar G.; Studden, W. J. Asymptotic Integrated Mean Square Error Using Least Squares and Bias Minimizing Splines. Ann. Statist. 8 (1980), no. 6, 1307--1325. doi:10.1214/aos/1176345203. http://projecteuclid.org/euclid.aos/1176345203.


Export citation