The Annals of Statistics

Two-Sided Tests and One-Sided Confidence Bounds

H. Finner

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Abstract

Based on the duality between tests and confidence sets we introduce a new method to derive one-sided confidence bounds following the rejection of a null hypothesis with two-sided alternatives. This method imputes that the experimenter is only interested in confidence bounds if the null hypothesis is rejected. Furthermore, we suppose that he is only interested in the direction and a lower confidence bound concerning the distance of the true parameter value to the parameter values in the null hypothesis. If the null hypothesis is rejected, the new one-sided confidence bounds are always not worse than the corresponding bounds of the two-sided confidence interval approach. If the true parameter is far away from the null hypothesis, the new bounds tend to be nearly equal to the corresponding one-sided confidence bounds with full confidence level $1 - \alpha$. The new method will be studied and illustrated in more detail in one-parameter exponential families and location families with unimodal Lebesgue densities, and, as an example where conditional tests are available, we consider the comparison of two Poission distributions. In case of the normal distribution with unknown variance we propose among others a modification of a procedure of Hodges and Lehmann. Here it may be surprising that there exist situations where the new method yields confidence bounds exactly matching the classical one-sided confidence bounds.

Article information

Source
Ann. Statist., Volume 22, Number 3 (1994), 1502-1516.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176325639

Digital Object Identifier
doi:10.1214/aos/1176325639

Mathematical Reviews number (MathSciNet)
MR1311986

Zentralblatt MATH identifier
0818.62021

JSTOR
links.jstor.org

Subjects
Primary: 62F03: Hypothesis testing
Secondary: 62F25: Tolerance and confidence regions

Keywords
Conditional inference composite $t$-test confidence interval confidence level directional errors exponential family location family normal distribution one-sided test Poisson distribution two-sided test unbiased uniformly most powerful unimodal

Citation

Finner, H. Two-Sided Tests and One-Sided Confidence Bounds. Ann. Statist. 22 (1994), no. 3, 1502--1516. doi:10.1214/aos/1176325639. https://projecteuclid.org/euclid.aos/1176325639


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