The Annals of Statistics

Sequential Confidence Regions in Inverse Regression Problems

Jiunn T. Hwang and Hung-Kung Liu

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Abstract

In inverse regression problems (or more generally, the estimation of ratios of regression parameters) and errors-in-variables models, it has been shown by Gleser and Hwang that the length of any confidence interval with positive confidence level is infinite with positive probability. Therefore the confidence sets derived using asymptotic theory, although having correct asymptotic coverage probability, typically have zero confidence level when the sample size is fixed. Is it possible to construct a sequential confidence interval with finite length and $1 - \alpha > 0$ confidence level? The answer is no for any finite stage sequential sampling. The answer is, however, yes for a fully sequential scheme, as demonstrated by Hwang and Liu. For the inverse regression problem, and more generally the set estimation of a ratio of regression parameters, we construct a $(1 - \alpha)$ confidence sequence. Applying such a confidence sequence, we can construct a $(1 - \alpha)$ sequential confidence interval with the length less than a prespecified quantity.

Article information

Source
Ann. Statist., Volume 18, Number 3 (1990), 1389-1399.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347756

Digital Object Identifier
doi:10.1214/aos/1176347756

Mathematical Reviews number (MathSciNet)
MR1062715

Zentralblatt MATH identifier
0709.62036

JSTOR
links.jstor.org

Subjects
Primary: 62F25: Tolerance and confidence regions
Secondary: 62L10: Sequential analysis 62H99: None of the above, but in this section 62F11

Keywords
Confidence level confidence region stopping rule calibration principal components analysis confidence sequences

Citation

Hwang, Jiunn T.; Liu, Hung-Kung. Sequential Confidence Regions in Inverse Regression Problems. Ann. Statist. 18 (1990), no. 3, 1389--1399. doi:10.1214/aos/1176347756. https://projecteuclid.org/euclid.aos/1176347756


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