Ordering extremes of exponentiated location-scale models with dependent and heterogeneous random samples
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Ordering extremes of exponentiated location-scale models with dependent and heterogeneous random samples Sangita Das1 · Suchandan Kayal1 Received: 15 March 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This paper is devoted to some ordering results for the largest and the smallest order statistics arising from dependent heterogeneous exponentiated location-scale random observations. We assume that the sets of observations are sharing a common or different Archimedean copula(s). Sufficient conditions for which the usual stochastic order and the reversed hazard rate order between the extreme order statistics hold are derived. Various numerical examples are provided for the illustration of the proposed results. Finally, some applications of the comparison results in engineering reliability and auction theory are presented. Keywords Archimedean copula · Order statistics · Majorization · Stochastic orders Mathematics Subject Classification 60E15 · 62G30 · 60K10
1 Introduction The order statistics play a significant role in several areas including risk management statistics, auction theory, reliability and many branches of applied probability. Consider a collection of n random variables {X 1 , . . . , X n }. We write X k:n for the kth smallest order statistic, k = 1, . . . , n. The ordered sample values X 1:n ≤ · · · ≤ X n:n are called the order statistics. Here, X 1:n and X n:n are respectively known as the smallest and the largest order statistics. It is known that a one-to-one correspondence holds between the lifetime of a k-out-of-n system and an order statistic. The lifetime of a k-out-of-n system is represented by the (n − k + 1)th order statistic. In literature, there are several applications of different order statistics. The maximum order statistic is
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Suchandan Kayal [email protected]; [email protected] Sangita Das [email protected]
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Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India
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S. Das, S. Kayal
useful in the study of floods. In planning, design and management of the hydraulic structures, estimation of rainfall for a certain period is of great interest to the scientists. This can be achieved by fitting probability distributions to the series of annual one-day maximum rainfall. Further, the maximum and the minimum order statistics can be used in reliability and survival studies since they respectively represent the survival times of the parallel and the series systems. Further, in actuarial science, X 1:n and X n:n can be respectively defined as the smallest and the largest potential losses bothered about a policy of multiple random risks. For more details on the order statistics, we refer to David and Nagaraja (2003). Different ordering results between two order statistics arising from various lifetime distributions have been studied by many authors. See for instance Barlow and Proschan (1975), Bartoszewicz (1986), Singh and Vijayasree (1991), Boland et al. (1998), Marshall et al. (2011), Balakrishnan and
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