Convolutions and generalization of logconcavity: Implications and applications

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

On unimodality of the lifetime distribution of coherent systems with failure-dependent component lifetimes

“…In our setting these just correspond to densities of single generalized order statistics. Unimodality in this case has been discussed in detail by Cramer et al (2004) and Bieniek (2007) (see also the recent results on strong unimodality in Alimohammadi et al (2016)).…”

“…A unifying framework for a number of models for ordered data including the usual order statistics is provided by generalized order statistics introduced by Kamps (1995) (see also Kamps (2016)). Results on unimodality in this larger model of generalized order statistics were presented in Cramer (2004), Cramer et al (2004), Chen et al (2009), Alimohammadi and Alamatsaz (2011), and Alimohammadi et al (2016).…”

Preservation of ILR and IFR aging classes in sums of dependent random variables

“…Example 3. Let us consider the sum of two dependent random variables X and Y having exponential and Rayleigh (Weibull) distributions with F(t) = exp(−t) and G(t) = exp(−t 2 ) for t ≥ 0 and with the FGM survival copula Ĉ defined in (6). Its partial derivative is…”

“…Example 4. Let us consider a random vector (X, Y ) with the FGM survival copula defined in (6). Let us assume that Y has a standard exponential distribution and that the survival function of X is…”

Formulation and Investigation of an Integral Equation for Characteristic Functions of Positive Random Variables

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