 Invariance, Optimality, and a 1-Observation Confidence Interval for a Normal MeanStephen Portnoy The American Statistician, 2019, vol. 73, issue 1, 10-15 Abstract: In a 1965 Decision Theory course at Stanford University, Charles Stein began a digression with “an amusing problem”: is there a proper confidence interval for the mean based on a single observation from a normal distribution with both mean and variance unknown? Stein introduced the interval with endpoints ± c|X| and showed indeed that for c large enough, the minimum coverage probability (over all values for the mean and variance) could be made arbitrarily near one. While the problem and coverage calculation were in the author’s hand-written notes from the course, there was no development of any optimality result for the interval. Here, the Hunt–Stein construction plus analysis based on special features of the problem provides a “minimax” rule in the sense that it minimizes the maximum expected length among all procedures with fixed coverage (or, equivalently, maximizes the minimal coverage among all procedures with a fixed expected length). The minimax rule is a mixture of two confidence procedures that are equivariant under scale and sign changes, and are uniformly better than the classroom example or the natural interval X ± c|X| . Date: 2019 Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:taf:amstat:v:73:y:2019:i:1:p:10-15 Ordering information: This journal article can be ordered from DOI: 10.1080/00031305.2017.1360796 Access Statistics for this article The American Statistician is currently edited by Eric Sampson More articles in The American Statistician from Taylor & Francis Journals |
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