Characterizations and modelling of multivariate lack of memory property. (English) Zbl 1100.62060
Summary: We establish characterizations of the multivariate lack of memory property in terms of the hazard gradient (whenever it exists), the survival function and the cumulative hazard function. Based on one of these characterizations we establish a method of generating bivariate lifetime distributions possessing the bivariate lack of memory property (BLMP) with specified marginals. It is observed that the marginal distributions have to satisfy certain conditions to be stated. The method generates absolutely continuous bivariate distributions as well as those containing a singular component. Bivariate exponential distributions due to F. Proschan and P. Sullo [F. Proschan and R. J. Serfling (eds.), Reliability and Biometry, 423–440 (1974)], J. E. Freund [J. Am. Stat. Assoc. 56, 971–977 (1961; Zbl 0106.13304)], H. W. Block and A. P. Basu [J. Am. Stat. Assoc. 69, 1031-1037 (1974; Zbl 0299.62027)] and A. W. Marshall and I. Olkin [J. Am. Stat. Assoc. 62, 30–44 (1967; Zbl 0147.38106)] are generated as particular cases, among others, using the proposed method.
Some other distributions generated using the method may be of practical importance. Shock models leading to bivariate distributions possessing BLMP are given. Some closure properties of a class of univariate failure rate functions that can generate distributions possessing BLMP and of the class of bivariate survival functions having the BLMP are studied.
Some other distributions generated using the method may be of practical importance. Shock models leading to bivariate distributions possessing BLMP are given. Some closure properties of a class of univariate failure rate functions that can generate distributions possessing BLMP and of the class of bivariate survival functions having the BLMP are studied.
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62N05 | Reliability and life testing |
Keywords:
multivariate lack of memory property; characterizations; modelling; hazard gradient; singular component; shock modelsReferences:
| [1] | Block HW, Basu AP (1974) A continuous bivariate exponential extension. J Am Stat Assoc 89:1091–1097 |
| [2] | Block HW (1977) Monotone hazard and failure rates for absolutely continuous multivariate distributions. Naval Res Logistic Quart 24:627–637 · Zbl 0383.62069 · doi:10.1002/nav.3800240410 |
| [3] | Ghurye SG, Marshall AW (1984) Shock processes with after effects and multivariate lack of memory property. J Appl Prob 21:786–801 · Zbl 0553.60078 · doi:10.2307/3213696 |
| [4] | Freund JE (1961) A bivariate extension of the exponential distribution. J Am Stat Assoc 56:971–977 · Zbl 0106.13304 · doi:10.2307/2282007 |
| [5] | Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Am Math Assoc 62:30–44 · Zbl 0147.38106 · doi:10.2307/2282907 |
| [6] | Marshall AW, Olkin I (1997) On adding a parameter to a distribution with special reference to exponential and Weibull models. Biometrika 84:641–652 · Zbl 0888.62012 · doi:10.1093/biomet/84.3.641 |
| [7] | Proschan F, Sullo P (1974) Estimating the parameters of a bivariate exponential distribution in several sampling situations. In: Proschan F, Serfling RJ (eds) Reliability and biometry. SIAM, Philadelfia, pp 423–440 |
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