Sequential Order Statistics

Abstract

Sequential order statistics have been introduced to model effects of component failures on the reliability of ‐out‐of‐ systems. Aside from this interpretation, they are related to several other models of ordered random variables such as Pfeifer's record values, generalized order statistics, and progressively type‐II censored order statistics.

Confidence bands for exponential distribution functions under progressive type-II censoring

“…More results on inference with SOSs can be found, for instance, in [29] and [30]. For recent works, see, e.g., [31] and [32].…”

Ordered and censored lifetime data in reliability: An illustrative review

“…Sequential order statistics X Ã 1:n , …, X Ã n:n based on cdfs F 1 ,…, F n have been introduced in Kamps (1995aKamps ( , 1995b) (see also Cramer, 2016;Cramer and Kamps, 2003) in order to model so-called sequential k-out-of-n systems (see Cramer and Kamps, 2001b) where failures of components may change the load imposed on remaining components in a k-out-of-n structure. In this regard, sequential order statistics can be considered as an extension of order statistics which are covered as a special case choosing F i = F for 1 ≤ i ≤ n. In the modeling process, the (conditional) hazard rate of a component lifetime after the ith failure is modeled by…”

“…For brevity, we present the simplest extensions of Type-I/Type-II hybrid censoring by presenting the corresponding test duration. In generalized Type-I hybrid censoring, one has T ¼ min max τ, X k:n f g,X r:n f g which has an unbounded test duration but the sample size D ▲ is bounded by k ≤ D ▲ ≤ r. Results on that censoring scheme, particularly on the exponential case, can be found in, for example, Chandrasekar et al (2004), Iliopoulos (2015), G orny and Cramer (2016), andCho et al (2015). In generalized Type-II hybrid censoring, the experimental time is given by T ¼ max min τ 1 , X r:n f g, τ 2 f gwhere the test duration is bounded by τ 2 .…”

Representations of component importance for coherent systems with exchangeable components

“…The formal definition of sequential order statistics can be found in [31] (see also [14, 16]). In particular, the resulting joint density of the sequential order statistics is given by for , and is zero elsewhere.…”

“…In order to illustrate the results, we examine in detail failure-dependent lifetimes based on the model of sequential order statistics (see [31]; see also [14, 15, 32]). Typically, the model is used to describe load-sharing scenarios in engineering applications (for this and further areas of application, see the additional references at the beginning of Section 3 and at the end of Section 4).…”