A note on improved variance bounds for certain bounded unimodal distributions
type: Item
instance of: scholarly article
title: A note on improved variance bounds for certain bounded unimodal distributions
zbMATH Open document ID: 0613.62019
DOI: 10.1016/0167-7152(86)90042-8
author: John W. jun. Seaman
author: Dean M. Young
author: Virgil R. Marco
published in: Statistics \& Probability Letters
publication date: 1986-01-01T00:00:00Z
review text: \textit{H. L. Gray} and \textit{P. L. Odell} [SIAM Rev. 9, 715-720 (1967; Zbl 0155.273)] have proved that no symmetric continuous unimodal density on the interval [a,b], with modes interior to (a,b), can have variance exceeding \((b-a)^ 2/12\). \textit{H. I. Jacobson} [Ann. Math. Stat. 40, 1746-1752 (1969; Zbl 0198.238)] has derived more general sufficient conditions for the application of this bound and also has shown that no unimodal distribution on [a,b] can have variance larger than \((b-a)^ 2/9\). \textit{P. L. Odell} and the first two authors [Stat. Probab. Lett. 3, 255-260 (1985; Zbl 0578.62022)] have presented even more general sufficient conditions for the smaller bound. In this note, we make use of a dispersion ordering to show that the previous conditions for the smaller bound are far too restrictive. Indeed, no continuous unimodal density f on [a,b], with f(a)\(\leq 1/(b-a)\) and f(b)\(\leq 1/(b-a)\), can have variance larger than \((b-a)^ 2/12\).
Mathematics Subject Classification ID: 62E99
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 3992628
zbMATH Keywords: improved variance bounds
zbMATH Keywords: variance inequality
zbMATH Keywords: symmetric continuous unimodal density
zbMATH Keywords: unimodal distribution
zbMATH Keywords: dispersion ordering
MaRDI profile type: Publication
full work available at URL: 0167-7152(86)90042-8
OpenAlex ID: W2095340512
cites work: On Least Favorable Density Functions
cites work: Q3234855
cites work: The Maximum Variance of Restricted Unimodal Distributions
cites work: Maximum variance unimodal distributions
cites work: Uniform conditional variability ordering of probability distributions
description: scientific article; zbMATH DE number 3992628