## The Annals of Statistics

- Ann. Statist.
- Volume 22, Number 3 (1994), 1328-1345.

### Simultaneous Confidence Bands for Linear Regression and Smoothing

Jiayang Sun and Clive R. Loader

#### Abstract

Suppose we observe $Y-i = f(x_i) + \varepsilon_i, i = 1, \ldots, n$. We wish to find approximate $1 - \alpha$ simultaneous confidence regions for $\{f(x), x \in \mathscr{X}\}$. Our regions will be centered around linear estimates $\hat{f}(x)$ of nonparametric or nonparametric $f(x)$. There is a large amount of previous work on this subject. Substantial restrictions have been usually placed on some or all of the dimensionality of $x,$ the class of functions $f$ that can be considered, the class of linear estimates $\hat{f}$ and the region $\mathscr{X}$. The method we present is an approximation to the tube formula dn can be used for multidimensional $x$ and a wide class of linear estimates. By considering the effect of bias we are able to relax assumptions on the class of functions $f$ which are considered. Simultaneous and numerical computations are used to illustrate the performance.

#### Article information

**Source**

Ann. Statist., Volume 22, Number 3 (1994), 1328-1345.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176325631

**Digital Object Identifier**

doi:10.1214/aos/1176325631

**Mathematical Reviews number (MathSciNet)**

MR1311978

**Zentralblatt MATH identifier**

0817.62057

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F25: Tolerance and confidence regions

Secondary: 60G15: Gaussian processes 62G07: Density estimation 62J05: Linear regression

**Keywords**

Linear smoother regression simultaneous confidence regions tube formula

#### Citation

Sun, Jiayang; Loader, Clive R. Simultaneous Confidence Bands for Linear Regression and Smoothing. Ann. Statist. 22 (1994), no. 3, 1328--1345. doi:10.1214/aos/1176325631. https://projecteuclid.org/euclid.aos/1176325631