Abstract
Two distinct methods for construction of some interesting new classes of multivariate probability densities are described and applied. As common results of both procedures three n-variate pdf classes are obtained. These classes are considered as generalizations of the class of univariate Weibullian, gamma, and multivariate normal pdfs. An example of an application of the obtained n-variate pdfs to the problem of modeling the reliability of multicomponent systems with stochastically dependent life-times of their components is given. Obtaining sequences over n = 2, 3, ... of consistent n-variate pdfs, that obey a relatively simple common pattern, for each n, allows us to extend some of the constructions from random vectors to discrete time stochastic processes. Application of one, so obtained, class of highly non-Markovian, but still sufficiently simple, stochastic processes for modeling maintenance of systems with repair, is presented. These models allow us to describe and analyze repaired systems with histories of all past repairs.
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B. C. Arnold and D. J. Strauss, “Bivariate distributions with exponential conditionals,” Journal of the American Statistical Association vol. 83 pp. 522–527, 1988.
B. C. Arnold and D. Strauss, “Bivariate distributions with conditionals in prescribed exponential families,” Journal of the Royal Statistical Society, Series B vol. 53 pp. 365–375, 1991.
B. C. Arnold, E. Castillo, and J. M. Sarabia, Conditionally Specified Distributions, Lecture Notes in Statistics—73, Springer: Berlin Heidelberg New York, 1992.
R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston: New York, 1975.
E. Castillo and J. Galambos, “Bivariate distributions with Weibull conditionals,” Analysis Mathematica vol. 16 pp. 3–9, 1990.
J. K. Filus, “On a type of dependencies between Weibull life times of system components,” Reliability Engineering and System Safety, vol. 31 no.3, 267–280, 1991.
J. K. Filus and L. Z. Filus, “A class of generalized multivariate normal densities,” Pakistan Journal of Statistics 1, vol. 16 pp. 11–32, 2000.
J. K. Filus and L. Z. Filus, “On some bivariate pseudonormal densities,” Pakistan Journal of Statistics, 1, vol. 17 pp. 1–19, 2001.
J. K. Filus and L. Z. Filus, “On the n-variate pseudonormal distributions,” In Preprint of the ICCS 7-th Conference, Lahore, Pakistan, Jan. 2–5, 2001.
J. K. Filus and L. Z. Filus, “Construction and analysis of n-variat pseudoweibullian and pseudogamma probability distributions,” Technical Report, No. 03-06-19 Department of Mathematics, Northeastern Illinois University, Chicago, IL. 60625, 2003.
J. K. Filus and L. Z. Filus, “Construction of some new stochastic processes,” Technical Report, No. 03-12-19, Department of Mathematics, Northeastern Illinois University, Chicago, IL. 60625, 2003.
J. K. Filus and L. Z. Filus, “On some new classes of multivariate probability distributions,” Pakistan Journal of Statistics vol. 22 pp. 21–42, 2006.
J. E. Freund, “A bivariate extension of the exponential distribution,” Journal of the American Statistical Association vol. 56 pp. 971–977, 1961.
Y. Hayakawa, “The construction of new bivariate exponential distributions from a Bayesian perspective,” Journal of the American Statistical Association vol. 89 pp. 1044–1049, 1994.
G. Heinrich and U. Jensen, “Parameter estimation for a bivariate life-distribution in reliability with multivariate extensions,” Metrika vol. 42 pp. 49–65, 1995.
S. Kotz, N. Balakrishnan, and N. L. Johnson, Continuous Multivariate Distributions, Volume 1, Second Edition, Wiley: New York, 2000.
D. V. Lindley and N. D. Singpurwalla, “Multivariate distributions for the life lengths of components of a system sharing a common environment,” Journal of Applied Probability vol. 23 pp. 418–431, 1986.
M. Loeve, Probability Theory, Volume 1, Fourth Edition, Springer- Verlag: Berlin Heidelberg New York, 1978.
J. Lu, “Weibull extensions of the Freund and Marshal–Olkin bivariate exponential models,” IEEE Transactions on Reliability vol. 38 no 5, pp. 615–619, 1989.
A. W. Marshall and I. Olkin, “A generalized bivariate exponential distribution,” Journal of Applied Probability vol. 4 pp. 291–303, 1967a.
A. W. Marshall and I. Olkin, “A multivariate exponential distribution,” Journal of the American Statistical Association vol. 62 pp. 33–44, 1967b.
J. Spreeuw, “Policy Values of Insurance Contracts on Two Lives with Dependent Lifetimes” at DeMoSTAFI Conference on Dependence Modeling: Statistical Theory and Applications in Finance and Insurance. Quebeck City, Canada, 20–22 May, 2004.
H. Wang, “A survey of maintenance policies of deteriorating system,” Journal of the Operational Research vol. 139 pp. 469–489, 2002.
X. Wang, G. C. Foliente, C. H. Wang, M. N. Nguyen, and R. H. Leicester, “Reliability-based Maintenance Planning of Timber Utility Poles Under Decay Attack” In ACSIM2002, Cairns, Australia, 2002.
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Filus, J., Filus, L. On New Multivariate Probability Distributions and Stochastic Processes with System Reliability and Maintenance Applications. Methodol Comput Appl Probab 9, 425–446 (2007). https://doi.org/10.1007/s11009-007-9026-6
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DOI: https://doi.org/10.1007/s11009-007-9026-6
Keywords
- Multivariate probability density (pdf)
- Stochastic process
- System reliability and maintenance modeling
- Actuary modeling