## The Annals of Statistics

### 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

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