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

Abstract

In this article, we establish some results concerning the univariate and multivariate likelihood ratio order of generalized order statistics and the special case of m-generalized order statistics and their associated conditional variables. These results, in addition to being new, also generalizes some of the known results in the literature. Finally, some applications of all these results are indicated.

Revisiting Multivariate Likelihood Ratio Ordering Results for Order Statistics

“…Additional results for conditional order statistics can be found in the recent works of Li and Zuo [26], Li and Chen [23], Asadi and Bairamov [2], Asadi [1], Li and Zhao [24], Khaledi and Shaked [20], Li and Zhao [25], and Sadegh [32]. Recently, Balakrishnan, Belzunce, Hami, and Khaledi [4] obtained several results for the likelihood ratio order of generalized order statistics. Again, it is possible to obtain additional results from general results in the setting of generalized order statistics; see, for example, Hu, Jin, and Khaledi [12], Xie and Hu [35], and Zhao and Balakrishnan [38] and Zhuang, Yao, and Hu [41].…”

“…. Following the ideas given in Balakrishnan et al [4], Theorem 3.1 can be used also to derive results for the comparison of conditional order statistics. , n. This result was proved in Belzunce, Franco, Ruiz, and Ruiz [6] and Lillo, Nanda, and Shaked [27] and now follows as a consequence of Theorem 3.1.…”

“…Examples of sublattices of interest in reliability are provided in [4]. Next, we recall some of them.…”

Convolution of Heterogeneous Bernoulli Random Variables and Some Applications

“…P roperties of conditional order statistics have been studied extensively in the literature; one may refer to Kochar and Xu [11] and Balakrishnan et al [1] and the references therein for further details. The result in Theorem 8 gives a new insight into the conditions for stochastically comparing the lifetimes of two k-out-of-n systems with heterogeneous components.…”

On a mixture representation of the conditional inactivity time of a coherent system

“…The result can be readily obtained from Theorem 3.7 of Balakrishnan et al (2010) or Theorem 3.2 of Belzunce et al (2011).…”