Convolutions and generalization of logconcavity: Implications and applications

Mahdi Alimohammadi, Mohammad Hossein Alamatsaz and Erhard Cramer

Naval Research Logistics (NRL), 2016, vol. 63, issue 2, 109-123

Abstract: Additive convolution of unimodal and α‐unimodal random variables are known as an old classic problem which has attracted the attention of many authors in theory and applied fields. Another type of convolution, called multiplicative convolution, is rather younger. In this article, we first focus on this newer concept and obtain several useful results in which the most important ones is that if fˆϕ is logconcave then so are Fˆϕ and F¯ˆϕ for some suitable increasing functions ϕ. This result contains ϕ(x)=x and ϕ(x)=ex as two more important special cases. Furthermore, one table including more applied distributions comparing logconcavity of f(x) and f(ex) and two comprehensive implications charts are provided. Then, these fundamental results are applied to aging properties, existence of moments and several kinds of ordered random variables. Multiplicative strong unimodality in the discrete case is also introduced and its properties are investigated. In the second part of the article, some refinements are made for additive convolutions. A remaining open problem is completed and a conjecture concerning convolution of discrete α‐unimodal distributions is settled. Then, we shall show that an existing result regarding convolution of symmetric discrete unimodal distributions is not correct and an easy alternative proof is presented. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 109–123, 2016

Date: 2016
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
https://doi.org/10.1002/nav.21679

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wly:navres:v:63:y:2016:i:2:p:109-123

Access Statistics for this article

More articles in Naval Research Logistics (NRL) from John Wiley & Sons
Bibliographic data for series maintained by Wiley Content Delivery ().