A Bi-Level Weibull Model with Applications to Two Ordered Events
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Published
Nov 16, 2020
Shuguang Song
Hanlin Liu
Mimi Zhang
Min Xie
Abstract
In this paper, we propose and study a new bivariate Weibull model, called Bi-levelWeibullModel, which arises when one failure occurs after the other. Under some specific regularity conditions, the reliability function of the second event can be above the reliability function of the first event, and is always above the reliability function of the transformed first event, which is a univariate Weibull random variable. This model is motivated by a common physical feature that arises fromseveral real applications. The two marginal distributions are a Weibull distribution and a generalized three-parameter Weibull mixture distribution. Some useful properties of the model are derived, and we also present the maximum likelihood estimation method. A real example is provided to illustrate the application of the model.
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Keywords
Bi-level Weibull distribution, Generalized three-parameter Weibull mixture, Maximum likelihood estimation
References
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Alzaatreh, A., Famoye, F., & Lee, C. (2013). Weibull-pareto distribution and its applications. Communications in Statistics - Theory and Methods, 42(9), 1673-1691.
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Bidram, H., Alamatsaz, M. H., & Nekoukhou, V. (2015). On an extension of the exponentiated weibull distribution. Communications in Statistics - Simulation and Computation, 44(6), 1389-1404.
Christer, A. H. (1999). Developments in delay time analysis for modeling plant maintenance. J. Opl. Res. Soc., 50, 1120-1137.
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Cooke, R. M., & Bedford, T. (2002). Reliability databases in perspective. IEEE Transactions on Reliability, 51(3), 294-310.
Franco, M., Balakrishnan, N., Kundu, D., & Vivo, J. M. (2014). Generalized mixtures of weibull components. TEST, 23(3), 515-535.
Franco, M., & Vivo, J.-M. (2009). Constraints for generalized mixtures of Weibull distributions with a common shape parameter. Statistics & Probability Letters, 79(15), 1724-1730.
Franco, M., Vivo, J. M., & Balakrishnan, N. (2011). Reliability properties of generalized mixtures of Weibull distributions with a common shape parameter. Journal of Statistical Planning and Inference, 141(8), 2600-2613.
Hanagal, D.(2006). BivariateWeibull regression model based on censored samples. Statistical Papers, 47(1), 137-147.
Hanagal, D. D. (2010). Modeling heterogeneity for bivariate survival data by theWeibull distribution. Statistical Papers, 51(4), 947-958.
Jiang, R., Zuo, M., & Li, H. (1999). Weibull and inverse Weibullmixturemodels allowing negative weights. Re- liability Engineering & System Safety, 66(3), 227-234.
Johnson, R. A., & Lu, W. (2007). Proof load designs for estimation of dependence in a bivariateWeibull model. Statistics & Probability Letters, 77(11), 1061-1069.
Jose, K. K., Ristic, M. M., & Joseph, A. (2011). Marshall- Olkin bivariate Weibull distributions and processes. Statistical Papers, 52(4), 789-798.
Karakoca, A., Erisoglu,U., & Erisoglu,M.(2015). A comparison of the parameter estimation methods for bimodal mixture weibull distribution with complete data. ournal of Applied Statistics, 42(7), 1472-1489.
Kojadinovic, I., & Yan, J.(2012). A non-parametric test of exchangeability for extreme-value and left-tail decreasing bivariate copulas. Scandinavian Journal of Statistics, 39(3), 480-496.
Kundu, D., & Gupta, R. D. (2011). Absolute continuous bivariate generalized exponential distribution. ASTA- Advances in Statistical Analysis, 95(2), 169-185.
Lai, C., Xie, M., & Murthy, D. (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52(1), 33-37.
Lu, J. C., & Bhattacharyya, G. K. (1990). Some new constructions of bivariate Weibull models. Annals of the Institute of StatisticalMathematics, 42, 543-559.
McCool, J. I. (2011). Software for weibull inference. Quality Engineering, 23, 253-264.
Nadar, M., & Kzlaslan, F. (2016). Estimation of reliability in a multicomponent stress-strength model based on a marshall-olkin bivariate weibull distribution. IEEE Transactions on Reliability, 65(1), 370-380.
Nourbakhsh, M., Mehrali, Y., Jamalizadeh, A., & Yari, G. (2015). On a selection weibull distribution. Communications in Statistics - Theory and Methods, 44(8), 1640-1652.
Panteleeva, O. V., Gonzlez, E. G., Huerta, H. V., & Alva, J. A. V. (2015). Identifiability and comparison of estimation methods on weibull mixture models. Communica- tions in Statistics - Simulation and Computation, 44(7), 1879-1900.
Saboor, A., Provost, S. B., & Ahmad,M.(2012). The moment generating function of a bivariate gamma-type distribution. Applied Mathematics and Computation, 218(24), 11911-11921.
Sauerbrei, W., & Royston, P. (1999). Building multivariable prognostic and diagnostic models: Transformation of the predictors by using fractional polynomials. Journal of the Royal Statistical Society: Series A, 162(1), 71-94.
Singla, N., Jain, K., & Sharma, S. K. (2012). The Beta Generalized Weibull distribution: Properties and applications. Reliability Engineering & System Safety, 102, 5-15.
Verrill, S. P., Evans, J. W., Kretschmann, D. E., & Hatfield, C. A. (2015). Asymptotically efficient estimation of a bivariate gaussianweibull distribution and an introduction to the associated pseudo-truncated weibull. Communications in Statistics - Theory and Methods, 44(14), 2957-2975.
Vrac, M., Billard, L., Diday, E., & Chedin, A.(2012). Copula analysis of mixture models. Computational Statistics, 27(3), 427-457.
Yeh, H.-C.(2012). Characterizations of the general multivariate weibull distributions. Communications in Statistics - Theory and Methods, 41(1), 76-87.
Alzaatreh, A., Famoye, F., & Lee, C. (2013). Weibull-pareto distribution and its applications. Communications in Statistics - Theory and Methods, 42(9), 1673-1691.
Balakrishnan, N., & Lai, C.-D. (2009). Continuous bivariate distributions (second ed.). Springer.
Bidram, H., Alamatsaz, M. H., & Nekoukhou, V. (2015). On an extension of the exponentiated weibull distribution. Communications in Statistics - Simulation and Computation, 44(6), 1389-1404.
Christer, A. H. (1999). Developments in delay time analysis for modeling plant maintenance. J. Opl. Res. Soc., 50, 1120-1137.
Cooke, R.M.(1996). The design of reliability data bases, part ii: Competing risk and data compression. Reliability Engineering & System Safety, 51, 209-223.
Cooke, R. M., & Bedford, T. (2002). Reliability databases in perspective. IEEE Transactions on Reliability, 51(3), 294-310.
Franco, M., Balakrishnan, N., Kundu, D., & Vivo, J. M. (2014). Generalized mixtures of weibull components. TEST, 23(3), 515-535.
Franco, M., & Vivo, J.-M. (2009). Constraints for generalized mixtures of Weibull distributions with a common shape parameter. Statistics & Probability Letters, 79(15), 1724-1730.
Franco, M., Vivo, J. M., & Balakrishnan, N. (2011). Reliability properties of generalized mixtures of Weibull distributions with a common shape parameter. Journal of Statistical Planning and Inference, 141(8), 2600-2613.
Hanagal, D.(2006). BivariateWeibull regression model based on censored samples. Statistical Papers, 47(1), 137-147.
Hanagal, D. D. (2010). Modeling heterogeneity for bivariate survival data by theWeibull distribution. Statistical Papers, 51(4), 947-958.
Jiang, R., Zuo, M., & Li, H. (1999). Weibull and inverse Weibullmixturemodels allowing negative weights. Re- liability Engineering & System Safety, 66(3), 227-234.
Johnson, R. A., & Lu, W. (2007). Proof load designs for estimation of dependence in a bivariateWeibull model. Statistics & Probability Letters, 77(11), 1061-1069.
Jose, K. K., Ristic, M. M., & Joseph, A. (2011). Marshall- Olkin bivariate Weibull distributions and processes. Statistical Papers, 52(4), 789-798.
Karakoca, A., Erisoglu,U., & Erisoglu,M.(2015). A comparison of the parameter estimation methods for bimodal mixture weibull distribution with complete data. ournal of Applied Statistics, 42(7), 1472-1489.
Kojadinovic, I., & Yan, J.(2012). A non-parametric test of exchangeability for extreme-value and left-tail decreasing bivariate copulas. Scandinavian Journal of Statistics, 39(3), 480-496.
Kundu, D., & Gupta, R. D. (2011). Absolute continuous bivariate generalized exponential distribution. ASTA- Advances in Statistical Analysis, 95(2), 169-185.
Lai, C., Xie, M., & Murthy, D. (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52(1), 33-37.
Lu, J. C., & Bhattacharyya, G. K. (1990). Some new constructions of bivariate Weibull models. Annals of the Institute of StatisticalMathematics, 42, 543-559.
McCool, J. I. (2011). Software for weibull inference. Quality Engineering, 23, 253-264.
Nadar, M., & Kzlaslan, F. (2016). Estimation of reliability in a multicomponent stress-strength model based on a marshall-olkin bivariate weibull distribution. IEEE Transactions on Reliability, 65(1), 370-380.
Nourbakhsh, M., Mehrali, Y., Jamalizadeh, A., & Yari, G. (2015). On a selection weibull distribution. Communications in Statistics - Theory and Methods, 44(8), 1640-1652.
Panteleeva, O. V., Gonzlez, E. G., Huerta, H. V., & Alva, J. A. V. (2015). Identifiability and comparison of estimation methods on weibull mixture models. Communica- tions in Statistics - Simulation and Computation, 44(7), 1879-1900.
Saboor, A., Provost, S. B., & Ahmad,M.(2012). The moment generating function of a bivariate gamma-type distribution. Applied Mathematics and Computation, 218(24), 11911-11921.
Sauerbrei, W., & Royston, P. (1999). Building multivariable prognostic and diagnostic models: Transformation of the predictors by using fractional polynomials. Journal of the Royal Statistical Society: Series A, 162(1), 71-94.
Singla, N., Jain, K., & Sharma, S. K. (2012). The Beta Generalized Weibull distribution: Properties and applications. Reliability Engineering & System Safety, 102, 5-15.
Verrill, S. P., Evans, J. W., Kretschmann, D. E., & Hatfield, C. A. (2015). Asymptotically efficient estimation of a bivariate gaussianweibull distribution and an introduction to the associated pseudo-truncated weibull. Communications in Statistics - Theory and Methods, 44(14), 2957-2975.
Vrac, M., Billard, L., Diday, E., & Chedin, A.(2012). Copula analysis of mixture models. Computational Statistics, 27(3), 427-457.
Yeh, H.-C.(2012). Characterizations of the general multivariate weibull distributions. Communications in Statistics - Theory and Methods, 41(1), 76-87.
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Technical Papers