Dynamic Multivariate Quantile Residual Life in Reliability Theory

M Shafaei Noughabi,A M Abouammoh,M Kayid

doi:10.1155/2018/1245656

M Shafaei Noughabi, A M Abouammoh + Show 1 more

Open Access

https://doi.org/10.1155/2018/1245656

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Abstract

We extend the univariate α-quantile residual life function to multivariate setting preserving its dynamic feature. Principal attributes of this function are derived and their relationship to the dynamic multivariate hazard rate function is discussed. A corresponding ordering, namely, α-quantile residual life order, for random vectors of lifetimes is introduced and studied. Based on the proposed ordering, a notion of positive dependency is presented. Finally, a discussion about conditions characterizing the class of decreasing multivariate α-quantile residual life functions is pointed out.

Highlights

For a random lifetime X, the α-quantile residual life function proposed by Haines and Singpurwalla (1974) describes the α-quantile of the well-known remaining lifetime of X given its survival at time x > 0
The condition investigated in the following theorem provides a simpler investigation of α-QRL-DF property which is similar to characterizations for mean residual lifetime (MRL)-DF, weakened by failure (WBF), supportive lifetimes (SL), hazard rate increase upon failure (HIF), and multivariate totally positive of order 2 (MTP2) presented in Shaked and Shanthikumar [11, 15]
The dynamic α-MQRL measure proposed in this paper is useful in both theoretical and applied aspects of reliability theory and survival analysis

Summary

Introduction

For a random lifetime X, the α-quantile residual life (αQRL) function proposed by Haines and Singpurwalla (1974) describes the α-quantile of the well-known remaining lifetime of X given its survival at time x > 0. The bivariate α-QRL (α-BQRL) function introduced by Shafaei Noughabi and Kayid [16] does not support histories of type (ii) This motivates us to extend the univariate α-quantile residual life function to multivariate setting preserving its dynamic feature. Mathematical Problems in Engineering function measures the α-quantile of the remaining lifetime conditioned on any possible history at this time. It can be regarded as a serious competitor for the multivariate MRL recommended by Shaked and Shanthikumar [15] and may even be preferred to that due to the comments of Schmittlein and Morrison [1]. To provide succinct notations, denote (x1, . . . , xi−1, t, xi+1, . . . , xn) by x⟨i; t⟩

Preliminaries

Dynamic Multivariate α-Quantile Residual Life

Multivariate α-Quantile Residual Life Order

Decreasing α-MQRL Class of Life Distributions

Conclusion