A note on reliability properties of k-record statistics

Raqab, Mohammad Z.; Amin, Wael A.

doi:10.1007/bf02717177

Cited by 9 publications

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“…Unimodality and strong unimodality of the distribution of ascendingly ordered random variables have been extensively studied (see, e.g., Alam [1], Huang & Ghosh [19], Raqab & Amin [29], Basak & Basak [7], Aliev [2], Cramer, Kamps & Rychlik [15], and Cramer [13]). Since generalized order statistics (GOSs), defined by Kamps ( [21,22]), contain all the models discussed by the above authors, recent results of Chen, Xie & Hu [12] and Alimohammadi & Alamatsaz [3] concerning GOSs have unified most of the previous findings.…”

Section: Introductionmentioning

confidence: 99%

SOME CONVEXITY PROPERTIES OF THE DISTRIBUTION OF LOWER k-RECORD VALUES WITH EXTENSIONS

Alimohammadi

Alamatsaz

Cramer³

2014

Prob. Eng. Inf. Sci.

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Unimodality and strong unimodality of the distribution of ascendingly ordered random variables have been extensively studied in the literature, whereas these properties have not received much attention in the case of descendingly ordered random variates. In this paper, we show that log concavity of the reversed hazard rate implies that of the density function. Using this fundamental result, we establish some convexity properties of such random variables. To do this, we first provide a counterexample showing that a claim of Basak & Basak [7] about the lower record values is not valid. Then, we provide conditions under which unimodality properties of the distribution of lower k-record values would hold. Finally, some extensions to dual generalized order statistics in both univariate and multivariate cases are discussed.

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Section: Introductionmentioning

confidence: 99%

SOME CONVEXITY PROPERTIES OF THE DISTRIBUTION OF LOWER k-RECORD VALUES WITH EXTENSIONS

Alimohammadi

Alamatsaz

Cramer³

2014

Prob. Eng. Inf. Sci.

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“…Unimodality and SU (logconcavity) properties of distribution of ordered random variables have received considerable attention in the literature (see, e.g., [4,18,20,26,28,34,47,52]). Since GOSs contain all models discussed by the above authors, the results of Chen et al [25] and Alimohammadi and Alamatsaz [10] concerning GOSs enfold most of the previous findings.…”

Section: Applications To Gossmentioning

confidence: 99%

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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A note on reliability properties of k-record statistics

Abstract: Record values, failure rate, IFR, DFR, log convex, log concave,

Cited by 9 publications

References 10 publications

SOME CONVEXITY PROPERTIES OF THE DISTRIBUTION OF LOWER k-RECORD VALUES WITH EXTENSIONS

SOME CONVEXITY PROPERTIES OF THE DISTRIBUTION OF LOWER k-RECORD VALUES WITH EXTENSIONS

Convolutions and generalization of logconcavity: Implications and applications

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