## On the residual lifelengths of the remaining components in an $$n - k+1$$ out of $$n$$ system.(English)Zbl 1141.62081

Summary: Suppose that a system consists of $$n$$ independent components and that the lifelength of the $$i$$ th component is a random variable $$X_i$$ $$(i=1,2,\dots, n)$$. For $$k\in \{1,2,\dots ,n - 1\}$$, denote by $$X_1^{(k)},X_2^{(k)},\dots ,X_{n-k}^{(k)}$$ the residual lifelengths of the remaining functioning components following the $$k$$ th failure in the system. We discuss the joint distribution of these exchangeable random variables. In addition, we identify the conditions sufficient to guarantee the independence of the residual lifelengths.

### MSC:

 62N05 Reliability and life testing 62E15 Exact distribution theory in statistics
Full Text:

### References:

 [1] Arnold, B.C., Pareto distributions, (1983), International Cooperative Publishing House Fairland · Zbl 1169.62307 [2] Arnold, B.C.; Balakrishnan, N.; Nagaraja, H.N., A first course in order statistics, (1992), Wiley New York · Zbl 0850.62008 [3] Asadi, M.; Bairamov, I., A note on the Mean residual life function of a parallel system, Communications in statistics. theory and methods, 34, 475-484, (2005) · Zbl 1062.62228 [4] Asadi, M.; Bairamov, I., The Mean residual life function of a $$k$$-out-of-$$n$$ structure at the system level, IEEE transactions on reliability, 55, 2, 314-318, (2006) [5] Bairamov, I.; Ahsanullah, M.; Akhundov, I., A residual life function of a system having parallel or series structures, Journal of statistical theory and applications, 1, 119-132, (2002) [6] Barlow, R.E.; Proschan, F., Statistical theory of reliability and life testing, probability models, (1975), Holt-Rinehart and Winston New York · Zbl 0379.62080 [7] Belzunce, F.; Franco, M.; Ruiz, J.M., On aging properties based on the residual life of $$k$$-out-of-$$n$$ systems, Probability in the engineering and informational sciences, 13, 193-199, (1999) · Zbl 0973.60103 [8] David, H.A., Order statistics, (1981), Wiley New York · Zbl 0223.62057 [9] David, H.A.; Nagaraja, H.N., Order statistics, (2003), Wiley-Interscience · Zbl 0905.62055 [10] Hall, W.J.; Wellner, J.A., Mean residual life, (), 169-184 · Zbl 0481.62078 [11] Kamps, U., A concept of generalized order statistics, (1995), Teubner Stuttgart · Zbl 0851.62035 [12] Meilijson, I., Limiting properties of the Mean residual life function, Annals of mathematical statistics, 42, 361-362, (1972) [13] Li, X.; Zhao, Peng, Some ageing properties of the residual life of $$k$$-out-of-$$n$$ systems, IEEE transactions on reliability, 55, 3, 535-541, (2006) [14] Li, X.; Zuo, M.J., On the behaviour of some new ageing properties based upon the residual life of $$k$$-out-of-$$n$$ systems, Journal of applied probability, 39, 426-433, (2002) · Zbl 1003.62089 [15] Oakes, D.; Dasu, T., A note on residual life, Biometrika, 77, 409-410, (1990) · Zbl 0713.62018 [16] Rao, C.R.; Shanbhag, D.N., Choquet – deny type functional equations with applications to stochastic models, (1994), Wiley Chichester · Zbl 0841.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.