Valid Confidence Intervals for $$\mu, \sigma $$ μ, σ When There Is Only One Observation Available

Anirban DasGupta () and Stephen Portnoy ()
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Anirban DasGupta: Purdue University
Stephen Portnoy: University of Illinois at Urbana-Champaign

Sankhya A: The Indian Journal of Statistics, 2024, vol. 86, issue 1, No 10, 173-184

Abstract: Abstract Portnoy (The American Statistician, 73:1, 10–15, 2019) considered the problem of constructing an optimal confidence interval for the mean based on a single observation $$\, X \sim \mathcal{{N}}(\mu , \, \sigma ^2) \,$$ X ∼ N ( μ , σ 2 ) . Here we extend this result to obtaining 1-sample confidence intervals for $$\, \sigma \,$$ σ and to cases of symmetric unimodal distributions and of distributions with compact support. Finally, we extend the multivariate result in Portnoy (The American Statistician, 73:1, 10–15, 2019) to allow a sample of size $$\, m \,$$ m from a multivariate normal distribution where m may be less than the dimension.

Keywords: Confidence interval/s; One sample value; Multivariate normal; Symmetric unimodal; Coverage probability; Non-central chi-square; Compact support; Primary- 62F10; secondary- 62C1 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13171-023-00338-2

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