On Marshall-Olkin type distribution with effect of shock magnitude. (English) Zbl 1329.62403
Summary: In classical Marshall-Olkin type shock models and their modifications a system of two or more components is subjected to shocks that arrive from different sources at random times and destroy the components of the system. With a distinctive approach to the Marshall-Olkin type shock model, we assume that if the magnitude of the shock exceeds some predefined threshold, then the component, which is subjected to this shock, is destroyed; otherwise it survives. More precisely, we assume that the shock time and the magnitude of the shock are dependent random variables with given bivariate distribution. This approach allows to meet requirements of many real life applications of shock models, where the magnitude of shocks is an important factor that should be taken into account. A new class of bivariate distributions, obtained in this work, involve the joint distributions of shock times and their magnitudes. Dependence properties of new bivariate distributions have been studied. For different examples of underlying bivariate distributions of lifetimes and shock magnitudes, the joint distributions of lifetimes of the components are investigated. The multivariate extension of the proposed model is also discussed.
MSC:
| 62N05 | Reliability and life testing |
| 62E10 | Characterization and structure theory of statistical distributions |
| 60E05 | Probability distributions: general theory |
| 60E15 | Inequalities; stochastic orderings |
References:
| [1] | Marshall, A. W.; Olkin, I., A multivariate exponential distribution, J. Amer. Statist. Assoc., 62, 30-44 (1967) · Zbl 0147.38106 |
| [2] | Ryu, K. W., An extention of Marshall and Olkin bivariate exponential distribution, J. Amer. Statist. Assoc., 88, 1458-1465 (1993) · Zbl 0799.62018 |
| [3] | Marshall, A. W.; Olkin, I., A new method of adding a parameter to a family of distributions with application to exponential and Weibull families, Biometrika, 84, 641-652 (1997) · Zbl 0888.62012 |
| [4] | Jayakumar, K.; Thomas, M., On a generalization to Marshall-Olkin scheme and its application to Burr type XII distribution, Statist. Papers, 49, 421-439 (2007) · Zbl 1310.62112 |
| [5] | Thomas, A.; Jose, K. K., Bivariate semi-Pareto minificaiton processes, Metrika, 59, 305-313 (2004) · Zbl 1147.62369 |
| [6] | Jose, K. K.; Ristic, M. M.; Joseph, A., Marshall-Olkin bivariate Weibull distributions and processes, Statist. Papers, 52, 789-798 (2011) · Zbl 1229.62066 |
| [7] | Ghitany, M. E.; Al-Hussaini, E. K.; Al-Jarallah, R. A., Marshall-Olkin extended Weibull distribution and its application to censored data, J. Appl. Stat., 32, 1025-1034 (2005) · Zbl 1121.62373 |
| [8] | Ghitany, M. E.; Al-Awadhi, F. A.; Alkhalfan, L. A., Marshall-Olkin extended Lomax distribution and its application to censored data, Comm. Statist. Theory Methods, 36, 1855-1866 (2007) · Zbl 1122.62081 |
| [9] | Li, X. H.; Pellerey, F., Generalized Marshall-Olkin distributions and related bivariate aging properties, J. Multivariate Anal., 102, 1399-1409 (2011) · Zbl 1221.60014 |
| [10] | Gupta, R. D.; Kundu, D., Generalized exponential distribution, Aust. N. Z. J. Stat., 41, 173-188 (1999) · Zbl 1007.62503 |
| [11] | Sarhan, A.; Balakrishnan, N., A new class of bivariate distributions and its mixture, J. Multivariate Anal., 98, 1508-1527 (2007) · Zbl 1116.62060 |
| [12] | Kundu, D.; Gupta, R. D., Bivariate generalized exponential distribution, J. Multivariate Anal., 100, 581-593 (2009) · Zbl 1169.62046 |
| [13] | Kundu, D.; Gupta, R. D., Modified Sarhan-Balakrishnan singular bivariate distribution, J. Statist. Plann. Inference, 140, 526-538 (2010) · Zbl 1177.62074 |
| [15] | Huang, Z. K.; Chau, K. W., A new image thresholding method based on Gaussian mixture model, Appl. Math. Comput., 205, 899-907 (2008) · Zbl 1152.68681 |
| [16] | Taormina, R.; Chau, K. W.; Sethi, R., Artificial neural network simulation of hourly groundwater levels in a coastal aquifer system of the Venice lagoon, Eng. Appl. Artif. Intell., 25, 1670-1676 (2012) |
| [17] | Wu, C. L.; Chau, K. W.; Li, Y. S., Predicting monthly streamflow using data-driven models coupled with data-preprocessing techniques, Water Resour. Res., 45 (2009), W08432 |
| [18] | Zhang, J.; Chau, K. W., Multilayer ensemble pruning via novel multi-sub-swarm particle Swarm optimization, J. UCS, 15, 840-858 (2009), 2009 |
| [19] | Cheng, C. T.; Chau, K. W.; Sun, Y.; Lin, J., Long-term prediction of discharges in Manwan reservoir using artificial neural network models, Lecture Notes in Comput. Sci., 3498, 1040-1045 (2005) · Zbl 1084.68579 |
| [20] | Chau, K. W., Application of a PSO-based neural network in analysis of outcomes of construction claims, Autom. Constr., 16, 642-646 (2007) |
| [21] | Gumbel, E. J., Bivariate exponential distributions, J. Amer. Statist. Assoc., 55, 698-707 (1960) · Zbl 0099.14501 |
| [22] | Balakrishnan, N.; Lai, C. D., Continuos Bivariate Distributions (2009), Springer: Springer Dordrecht · Zbl 1267.62028 |
| [23] | Bairamov, I.; Kotz, S., Dependence structure and symmetry of Huang-Kotz FGM distributions and their extensions, Metrika, 56, 1, 55-72 (2002) · Zbl 1433.62044 |
| [24] | Bairamov, I.; Kotz, S.; Bekci, M., New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics, J. Appl. Stat., 28, 5, 521-536 (2001) · Zbl 0991.62032 |
| [25] | Lai, C. D., Morgenstern’s bivariate distribution and its application to point process, J. Math. Anal. Appl., 65, 247-256 (1978) · Zbl 0388.60052 |
| [26] | Nadarajah, S.; Kotz, S., Performance measures for some bivariate Pareto distributions, Int. J. Gen. Syst., 35, 387-393 (2006) · Zbl 1140.93479 |
| [27] | Hanagal, D. D., Note on estimation of reliability under bivariate Pareto stress-strength model, Statist. Papers, 38, 453-459 (1997) · Zbl 0911.62091 |
| [28] | Navarro, J.; Ruiz, J. M.; Sandoval, C. J., Properties of systems with two exchangeable Pareto components, Statist. Papers, 49, 177-190 (2008) · Zbl 1168.62387 |
| [29] | Papadakis, E. N.; Tsionas, E. G., Multivariate Pareto distributions: inference and financial applications, Comm. Statist. Theory Methods, 39, 1013-1025 (2010) · Zbl 1189.91228 |
| [30] | Nayak, T. K., Multivariate Lomax distribution: properties and usefulness in reliability theory, J. Appl. Probab., 24, 170-177 (1987) · Zbl 0615.60082 |
| [31] | Sankaran, P. G.; Nair, U. N., A bivariate Pareto model and its applications to reliability, Nav. Res. Logist., 40, 1013-1020 (1993) · Zbl 0801.90049 |
| [32] | Lai, C. D.; Xie, M.; Bairamov, I. G., Dependence and ageing properties of bivariate Lomax distribution, (Hayakawa, Y.; Irony, T.; Xie, M., System and Bayesian Reliability: Essays in Honor of Prof. R.E. Barlow on His 70th Birthday (2001), World Scientific: World Scientific Singapore), 243-256 |
| [33] | Nelsen, R. B., An Introduction to Copulas (2006), Springer: Springer New York · Zbl 1152.62030 |
| [34] | Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007), Springer: Springer New York · Zbl 1111.62016 |
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