## The Annals of Statistics

- Ann. Statist.
- Volume 16, Number 2 (1988), 713-732.

### Approximation of Least Squares Regression on Nested Subspaces

#### Abstract

For a regression model $y_i = \theta(x_i) + \varepsilon_i$, the unknown function $\theta$ is estimated by least squares on a subspace $\Lambda_m = \operatorname{span}\{\psi_1, \psi, \cdots, \psi_m\}$, where the basis functions $\psi_i$ are predetermined and $m$ is varied. Assuming that the design is suitably approximated by an asymptotic design measure, a general method is presented for approximating the bias and variance in a scale of Hilbertian norms natural to the problem. The general theory is illustrated with two examples: truncated Fourier series regression and polynomial regression. For these examples, we give rates of convergence of derivative estimates in (weighted) $L_2$ norms and establish consistency in supremum norm.

#### Article information

**Source**

Ann. Statist., Volume 16, Number 2 (1988), 713-732.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350830

**Digital Object Identifier**

doi:10.1214/aos/1176350830

**Mathematical Reviews number (MathSciNet)**

MR947572

**Zentralblatt MATH identifier**

0669.62047

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J05: Linear regression

Secondary: 62F12: Asymptotic properties of estimators 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10}

**Keywords**

Regression nonparametric regression bias approximation polynomial regression model selection rates of convergence orthogonal polynomials

#### Citation

Cox, Dennis D. Approximation of Least Squares Regression on Nested Subspaces. Ann. Statist. 16 (1988), no. 2, 713--732. doi:10.1214/aos/1176350830. https://projecteuclid.org/euclid.aos/1176350830