Some comparisons between generalized order statistics

Abstract

Some stochastic comparisons of generalized order statistics under the right spread order, the location independent riskier order and the total time transform order are investigated in this paper. The underlying distributions and parameters on which generalized order statistics are based are also surveyed to obtain the conditions for increasing the expectations of spacings between the first two generalized order statistics and between the last two generalized order statistics. §1 IntroductionThe concept of generalized order statistics was introduced by Kamps [1-2] as a unified approach to a variety of models of ordered random variables (rv's). Choosing the parameters approximately, several other models of ordered rv's, such as ordinary order statistics, sequential order statistics, k-record values, etc, are seen to be the particular cases. Generalized order statistics (GOS) have been of interest in the past ten years because they are more flexible in reliability theory, statistical modeling and inference, see [3][4][5][6][7].A natural question for GOSs is whether it possesses analogical properties of ordinary order statistics. Readers can refer to [8][9][10][11][12][13] to obtain stochastic comparisons of GOSs and their spacings by analogy with ordinary order statistics. For the aging preservations of spacings of GOSs, readers can refer to [13][14]. Recently, Wang and Li [15] exhibited preservations of ordinary order statistics under some stochastic orders; Qiu and Shen [16] found that expectations of spacings between the first two ordinary order statistics and between the last two ordinary order statistics will increase under suitable conditions.The first purpose of this paper, after recalling the definitions of generalized order statistics, some stochastic orders and three useful lemmas in §2, is to build some stochastic comparisons of generalized order statistics under the right spread order, the location independent riskier

On sufficient conditions for the comparison in the excess wealth order and spacings

“…Our purpose is to compare the size of simple spacings of GOSs based on two independent distributions F and G when the underlying random variables X and Y fail to be ordered in the dispersive order. One possibility is to use the following result of Qiu and Wang (2007):…”

Univariate and Multivariate Likelihood Ratio Ordering of Generalized Order Statistics and Associated Conditional Variables

“…Stochastic comparisons of GOSs have been discussed rather extensively by Franco, Ruiz, and Ruiz [10], Belzunce, Mercade, and Ruiz [9], Khaledi [16], Hu and Zhuang [12,13], Khaledi and Kochar [18], Qiu and Wu [29], and Xie and Hu [35]. The stochastic comparison of conditional generalized order statistics has also been considered as a natural extension of the corresponding results on conditional order statistics and record values.…”

Some Results on Stochastic Orderings of Generalized Order Statistics and Spacings

“…Because of generality of generalized order statistics, several results on stochastic comparisons of ordinary order statistics and the corresponding sample spacings have been extended to generalized order statistics and their spacings. Interested readers may refer to [17], [9], [20], [25], [16], [34], [39], [40] and [10] for a brief review. In this paper, we follow this idea and try to generalize some ordering results in the literature from ordinary order statistics and their sample spacings to generalized order statistics and the corresponding spacings.…”