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

Published online by Cambridge University Press:  17 March 2014

Mahdi Alimohammadi ,
Mohammad Hossein Alamatsaz  and
Erhard Cramer
Mahdi Alimohammadi
Affiliation:
Department of Statistics, University of Isfahan, Isfahan, 81746-73441, Iran
Mohammad Hossein Alamatsaz
Affiliation:
Department of Statistics, University of Isfahan, Isfahan, 81746-73441, Iran
Erhard Cramer
Affiliation:
Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany. Email: erhard.cramer@rwth-aachen.de
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Abstract

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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.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 3 , July 2014 , pp. 389 - 399
Copyright
Copyright © Cambridge University Press 2014 

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