## The Annals of Statistics

- Ann. Statist.
- Volume 21, Number 4 (1993), 1780-1791.

### Bootstrap Confidence Regions for Functional Relationships in Errors-in- Variables Models

#### Abstract

We suggest bootstrap methods for constructing confidence bands (and intervals) for an unknown linear functional relationship in an errors-invariables model. It is assumed that the ratio of error variances is known to lie within an interval $\Lambda = \lbrack\lambda_1, \lambda_2\rbrack$. A confidence band is constructed for the range of possible linear relationships when $\lambda \in \Lambda$. Meaningful results are obtained even in the extreme case $\Lambda = \lbrack 0, \infty\rbrack$, which corresponds to no assumption being made about $\Lambda$. The bootstrap bands have several interesting features, which include the following: (i) the bands do not shrink to a line as $n \rightarrow \infty$, unless $\Lambda$ is a singleton (i.e., $\lambda_1 = \lambda_2)$; (ii) percentile-$t$ versions of the bands enjoy only first-order coverage accuracy, not the second-order accuracy normally found in simpler statistical problems.

#### Article information

**Source**

Ann. Statist., Volume 21, Number 4 (1993), 1780-1791.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349397

**Digital Object Identifier**

doi:10.1214/aos/1176349397

**Mathematical Reviews number (MathSciNet)**

MR1245768

**Zentralblatt MATH identifier**

0789.62035

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J05: Linear regression

Secondary: 62G15: Tolerance and confidence regions

**Keywords**

Bootstrap confidence band confidence interval Edgeworth expansion errors-in-variables model functional relationship percentile method percentile-$t$ method structural relationship

#### Citation

Booth, James G.; Hall, Peter. Bootstrap Confidence Regions for Functional Relationships in Errors-in- Variables Models. Ann. Statist. 21 (1993), no. 4, 1780--1791. doi:10.1214/aos/1176349397. https://projecteuclid.org/euclid.aos/1176349397