A note on improved variance bounds for certain bounded unimodal distributions
\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 \((b-a)^ 2/12\).
scientific article; zbMATH DE number 3992628