Abstract
There are several functions in reliability theory used to describe the patterns of failure in different mechanisms or systems as a function of age. The functional forms of many of these concepts characterize the life distribution and therefore enable the identification of the appropriate model. In this chapter, we discuss these basic concepts, first using the distribution function approach and then introduce their analogues in terms of quantile functions. Various important concepts introduced here include the hazard rate, mean residual life, variance residual life, percentile residual life, coefficient of variation of residual life, and their counterparts in reversed time. The expressions for all these functions for standard life distributions are given in the form of tables to facilitate easy reference. Formulas for the determination of the distribution from these functions, their characteristic properties and characterization theorems for different life distributions by relationships between various functions are reviewed. Many of the quantile functions in the literature do not have closed-form expressions for their distributions, and they have to be evaluated numerically. This renders analytic manipulation of these reliability functions based on the distribution function rather difficult. Accordingly, we introduce equivalent definitions and properties of the traditional concepts in terms of quantile functions. This leads to hazard quantile function, mean residual quantile function and so on. The interrelationships between these functions are presented along with characterizations. Various examples given in the sequel illustrate how the quantile based reliability functions can be found directly from the quantile functions of life distributions. Expressions of such functions for standard life distributions can also be read from the tables provided in each case.
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References
Abouammoh, A.M., Kanjo, A., Khalique, A.: On aspects of variance residual life distributions. Microelectron. Reliab. 30, 751–760 (1990)
Abraham, B., Nair, N.U.: On characterizing mixtures of life distributions. Stat. Paper. 42, 387–393 (2001)
Adamidis, K., Dimitrakopoulou, T., Loukas, S.: On an extension of the exponential geometric distribution. Stat. Probab. Lett. 73, 259–269 (2005)
Adatia, A., Law, A.G., Wang, Q.: Characterization of mixture of gamma distributions via conditional moments. Comm. Stat. Theor. Meth. 20, 1937–1949 (1991)
Arnold, B.C., Brockett, P.L.: When does the β-th percentile residual life function determine the distribution. Oper. Res. 31, 391–396 (1983)
Block, H.W., Savits, T.H., Singh, H.: The reversed hazard rate function. Probab. Eng. Inform. Sci. 12, 69–90 (1998)
Consul, P.C.: Some characterizations of the exponential class of distributions. IEEE Trans. Reliab. 40, 290–295 (1995)
Dallas, A.C.: A characterization using conditional variances. Metrika 28, 151–153 (1981)
El-Arishi, S.: A conditional probability characterization of some discrete probability distributions. Stat. Paper. 46, 31–45 (2005)
Finkelstein, M.S.: On the reversed hazard rate. Reliab. Eng. Syst. Saf. 78, 71–75 (2002)
Ghitany, M.E., El-Saidi, M.A., Khalil, Z.: Characterization of a general class of life testing models. J. Appl. Probab. 32, 548–553 (1995)
Gilchrist, W.G.: Statistical Modelling with Quantile Functions. Chapman and Hall/CRC Press, Boca Raton (2000)
Gore, A.P., Paranjpe, S.A., Rajarshi, M.B., Gadgul, M.: Some methods of summarising survivorship in nonstandard situations. Biometrical J. 28, 577–586 (1986)
Guess, F., Proschan, F.: Mean residual life: Theory and Applications. In: Krishnaiah, P.R., Rao, C.R. (eds.) Handbook of Statistics, vol. 7, pp. 215–224. North-Holland, Amsterdam (1988)
Gupta, R.C.: On the monotonic properties of residual variance and their applications in reliability. J. Stat. Plann. Infer. 16, 329–335 (1987)
Gupta, R.C., Bradley, D.M.: Representing mean residual life in terms of failure rate. Math. Comput. Model. 1, 1–10 (2003)
Gupta, R.C., Kirmani, S.N.U.A.: Residual coefficient of variation and some characterization results. J. Stat. Plann. Infer. 91, 23–31 (2000)
Gupta, R.C., Kirmani, S.N.U.A.: Some characterizations of distributions by functions of failure rate and mean residual life. Comm. Stat. Theor. Meth. 33, 3115–3131 (2004)
Gupta, R.C., Kirmani, S.N.U.A., Launer, R.L.: On life distributions having monotone residual variance. Probab. Eng. Inform. Sci. 1, 299–307 (1987)
Gupta, R.C., Langford, E.S.: On the determination of a distribution by its median residual life function: a functional equation. J. Appl. Probab. 21, 120–128 (1984)
Haines, A.L., Singpurwalla, N.D.: Some contributions to the stochastic characterization of wear. In: Proschan, F., Serfling, R.J. (eds.) Reliability and Biometry: Statistical Analysis of Lifelength, pp. 47–80. Society for Industrial and Applied Mathematics, Philadelphia (1974)
Hjorth, U.: A reliability distribution with increasing, decreasing, constant and bathtub shaped failure rates. Technometrics 22, 99–107 (1980)
Jaisingh, L.R., Kolarik, W.J., Dey, D.K.: A flexible bathtub hazard model for nonrepairable systems with uncensored data. Microelectron. Reliab. 27, 87–103 (1987)
Jiang, R., Murthy, D.N.P.: Two sectional models involving three Weibull distributions. Qual. Reliab. Eng. Int. 13, 83–96 (1997)
Joe, H.: Characterization of life distributions from residual lifetimes. Ann. Inst. Stat. Math. 37, 165–172 (1985)
Keilson, J., Sumita, U.: Uniform stochastic ordering and related inequalities. Can. J. Stat. 15, 63–69 (1982)
Koicheva, M.: A characterization of gamma distribution in terms of conditional moments. Appl. Math. 38, 19–22 (1993)
Kundu, D., Nanda, A.K.: Some reliability properties of the inactivity time. Comm. Stat. Theor. Meth. 39, 899–911 (2010)
Lai, C.D., Xie, M.: Stochastic Ageing and Dependence for Reliability. Springer, New York (2006)
Launer, R.L.: Graphical techniques for analysing failure time data with percentile residual life. IEEE Trans. Reliab. 42, 71–80 (1983)
Launer, R.L.: Inequalities for NBUE and NWUE life distributions. Oper. Res. 32, 660–667 (1984)
Lillo, R.E.: On the median residual lifetime and its aging properties: A characterization theorem and applications. Nav. Res. Logist. 52, 370–380 (2005)
Lin, G.D.: On the characterization of life distributions by percentile residual lifetimes. Sankhyā 71, 64–72 (2009)
Marshall, A.W., Olkin, I.: Life Distributions. Springer, New York (2007)
Mukherjee, S.P., Roy, D.: Some characterizations of the exponential and related life distributions. Calcutta Stat. Assoc. Bull. 35, 189–197 (1986)
Muth, E.J.: Reliability models with positive memory derived from mean residual life function. In: Tsokos, C.P., Shimi, I.N. (eds.) Theory and Application of Reliability, pp. 401–435. Academic, Boston (1977)
Nair, N.U., Asha, G.: Characterizations using failure and reversed failure rates. J. Indian Soc. Probab. Stat. 8, 45–56 (2004)
Nair, N.U., Sankaran, P.G.: Characterization of the Pearson family of distributions. IEEE Trans. Reliab. 40, 75–77 (1991)
Nair, N.U., Sankaran, P.G.: Characterization of multivariate life distributions. J. Multivariate Anal. 99, 2096–2107 (2008)
Nair, N.U., Sankaran, P.G.: Quantile-based reliability analysis. Comm. Stat. Theor. Meth. 38, 222–232 (2009)
Nair, N.U., Sankaran, P.G.: Some results on an additive hazard model. Metrika 75, 389–402 (2010)
Nair, N.U., Sudheesh, K.K.: Characterization of continuous distributions by variance bound and its implications to reliability modelling and catastrophe theory. Comm. Stat. Theor. Meth. 35, 1189–1199 (2006)
Nair, N.U., Sudheesh, K.K.: Some results on lower variance bounds useful in reliability modelling and estimation. Ann. Inst. Stat. Math. 60, 591–603 (2008)
Nair, N.U., Sudheesh, K.K.: Characterization of continuous distributions by properties of conditional variance. Stat. Methodol. 7, 30–40 (2010)
Nair, N.U., Vineshkumar, B.: Reversed percentile residual life and related concepts. J. Kor. Stat. Soc. 40, 85–92 (2010)
Nanda, A.K., Bhattacharjee, S., Alam, S.S.: Properties of proportional mean residual life model. Stat. Probab. Lett 76, 880–890 (2006)
Navarro, J., Franco, M., Ruiz, J.M.: Characterization through moments of residual life and conditional spacing. Sankhyā Ser. A 60, 36–48 (1998)
Osaki, S., Li, X.: Characterizations of the gamma and negative binomial distributions. IEEE Trans. Reliab. 37, 379–382 (1988)
Sankaran, P.G., Nair, N.U., Sindhu, T.K.: A generalized Pearson system useful in reliability analysis. Stat. Paper. 44, 125–130 (2003)
Schmittlein, D.C., Morrison, D.G.: The median residual lifetime–a characterization problem and an application. Oper. Res. 29, 392–399 (1981)
Song, J.K., Cho, G.Y.: A note on percentile residual life. Sankhyā 57, 333–335 (1995)
Wang, F.K.: A new model with bathtub-shaped hazard rate using an additive Burr XII distribution. Reliab. Eng. Syst. Saf. 70, 305–312 (2000)
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Nair, N.U., Sankaran, P.G., Balakrishnan, N. (2013). Quantile-Based Reliability Concepts. In: Quantile-Based Reliability Analysis. Statistics for Industry and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-8361-0_2
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