Some new results about shifted hazard and shifted likelihood ratio orders

Abstract

Shifted stochastic orders are a useful tool for establishing interesting inequalities. Two such stochastic orders are the up hazard rate order and the up likelihood ratio order. In this article, we give some results on the preservation of the above orderings between the components under the formation of coherent systems with different structures. Both the cases when components either identically distributed or not necessary identically distributed are discussed. Finally, we provide some applications to Poisson and non-homogeneous Poisson shock models.

Applications of TP2 Functions in Theory of Stochastic Orders: A Review of some Useful Results

“…124), one realizes that Theorem 4.1 can, in fact, be proved under weaker assumptions detailed below (see, for example, proof of Theorem 1.1 in Hu and Zhu (2001)). For this purpose, we first introduce some shifted stochastic orders (that are stronger than their counterparts in Definition 2.1) which have been studied in the literature (see, for example, Shanthikumar and Yao (1986), Nakai (1995), Brown and Shanthikumar (1998), Lillo et al (2000), Lillo et al (2001), Di Crescenzo and Longobardi (2001), Hu and Zhu (2001), Belzunce et al (2002), Nanda et al (2006), and Aboukalam and Kayid (2007)).…”

Some stochastic comparison results for frailty and resilience models

“…One such specific instance is that we can compare the reliability of an used device and a new device using the shifted stochastic orders. For more discussion on those shifted orders including their applications and preservations properties, we refer to boukalam and Kayid (2007), Naqvi et al (2021), Kayid et al (2017b) and references therein.…”

Some ranking indexes of stochastic orders and their applications

“…They have compared them and obtained some basic and closure properties of them and have shown how those can be used for stochastic comparisons of order statistics. Recently, Aboukalam and Kayid [1] obtained some new results about shifted hazard and shifted likelihood ratio orders. In this paper we recall the proportional state of stochastic orders and the shifted version of them and so obtained some applications of proportional Hazard Rate order.…”