Discrete strong unimodality of order statistics

Alimohammadi, Mahdi; Alamatsaz, Mohammad Hossein; Cramer, Erhard

doi:10.1016/j.spl.2015.04.019

Cited by 7 publications

(9 citation statements)

References 18 publications

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“…This is equivalent to concavity of

F ˆ ϕ

. Since increasing concave functions are logconcave we have the result.Remark (i) Sengupta and Nanda presented Theorem for

ϕ (x) = x

. (ii) Recently, Alimohammadi et al obtained the discrete version of Theorems , , and for

ϕ (x) = x

(where x is integer). They gave Theorem only for the case when

- \infty < L

and

U = \infty

. …”

Section: Multiplicative Convolutions Of Unimodal Random Variablesmentioning

confidence: 74%

“…Using implications i and a in Fig. 1 as well as mathematical induction, according to the conditions in (i) and (ii) the integrands in (12) and (13) are both seen to be logconcave. Thus, by Theorem 2.6 the integral g r (F (x)) is logconcave.…”

Section: Applications To Gossmentioning

confidence: 99%

“…As pointed out in the introduction, the unimodality property is not preserved under convolutions. There exist several counterexamples in the continuous case (see [31], p. [11][12][13] and also one counterexample in the discrete case (see [31], p. 108). However, they are all somewhat complicated.…”

Section: Some Useful Counterexamplesmentioning

confidence: 99%

“…Recently, Alimohammadi et al completed these results in the continuous case by obtaining some useful results on the reversed hazard rate. Also, for the first time, Alimohammadi et al considered logconcavity of order statistics for discrete random variables.…”

Section: Multiplicative Convolutions Of Unimodal Random Variablesmentioning

confidence: 99%

“…Since increasing concave functions are logconcave we have the result. Alimohammadi et al[12] obtained the discrete version of Theorems 2.6, 2.10, and 2.11 for φ(x) = x (where x is integer). They gave Theorem 2.10 only for the case when −∞ < L and U = ∞.…”

mentioning

confidence: 99%

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Convolutions and generalization of logconcavity: Implications and applications

Alimohammadi

Alamatsaz

Cramer

2016

Naval Research Logistics

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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

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“…This is equivalent to concavity of