A generalized unimodality
- 1 April 1970
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 7 (1) , 21-34
- https://doi.org/10.2307/3212145
Abstract
Khintchine (1938) showed that a real random variable Z has a unimodal distribution with mode at 0 iff Z ~ U X (that is, Z is distributed like U X), where U is uniform on [0, 1] and U and X are independent. Isii ((1958), page 173) defines a modified Stieltjes transform of a distribution function F for w complex thus: Apparently unaware of Khintchine's work, he proved (pages 179-180) that F is unimodal with mode at 0 iff there is a distribution function Φ for which . The equivalence of Khintchine's and Isii's results is made vivid by a proof (due to L. A. Shepp) in the next section.Keywords
This publication has 2 references indexed in Scilit:
- On the Composition of Unimodal DistributionsTheory of Probability and Its Applications, 1956
- The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalitiesProceedings of the American Mathematical Society, 1955