Abstract
To compare the variability of two random variables, we can use a partial order relation defined on a distribution class, which contains the anti-symmetry. Recently, Nair et al. studied the properties of total time on test (TTT) transforms of order n and examined their applications in reliability analysis. Based on the TTT transform functions of order n, they proposed a new stochastic order, the TTT transform ordering of order n (TTT-n), and discussed the implications of order TTT-n. The aim of the present study is to consider the closure and reversed closure of the TTT-n ordering. We examine some characterizations of the TTT-n ordering, and obtain the closure and reversed closure properties of this new stochastic order under several reliability operations. Preservation results of this order in several stochastic models are investigated. The closure and reversed closure properties of the TTT-n ordering for coherent systems with dependent and identically distributed components are also obtained.
Highlights
The aim of the present study is to consider the closure and reversed closure of the time on test (TTT)-n ordering
As applications of a main result Theorem 1, in Section 4, we examine the preservation of the TTT transform ordering of order n (TTT-n) ordering in several stochastic models
The concept of TTT transform is of great significance in engineering and technologies, experiment science, and other related scientific research fields
Summary
“Symmetry” is usually used to refer to an object that is invariant under some transformations; including translation, reflection, etc. (for example, Zee [1]). The concept of total time on test (TTT) transforms is of significant importance for its applications in different study fields such as reliability theory and economics. By using the TTT transform functions, Kochar et al [20] established the following TTT transform ordering and gave this stochastic order a careful study. Based on the GTTT transform functions, they defined the following new stochastic order. Franco-Pereira and Shaked [19] studied the TTT transform and the decreasing percentile residual life aging notion. On the basis of the work of Nair and Sankaran [23], they added two characterizations of the decreasing percentile residual life of order α (DPRL(α)) aging notion in terms of the TTT function, and in terms of the observed. Assume that all random variables involved are absolutely continuous and non-negative, and that all integrals appeared are finite and all ratios are well defined whenever written
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