A Bayesian Inference Approach for Bivariate Weibull Distributions Derived from Roy and Morgenstern Methods

  • Ricardo Puziol de Oliveira State University of Maring
  • Marcos Vinicius de Oliveira Peres University of São Paulo
  • Milene Regina dos Santos University of São Paulo
  • Edson Zangiacomi Martinez University of São Paulo
  • Jorge Aberto Achcar University of São Paulo
Keywords: Bayesian analysis, bivariate lifetime model, Morgenstern family, stress-strength property


Bivariate lifetime distributions are of great importance in studies related to interdependent components, especially in engineering applications. In this paper, we introduce two bivariate lifetime assuming three- parameter Weibull marginal distributions. Some characteristics of the proposed distributions as the joint survival function, hazard rate function, cross factorial moment and stress-strength parameter are also derived. The inferences for the parameters or even functions of the parameters of the models are obtained under a Bayesian approach. An extensive numerical application using simulated data is carried out to evaluate the accuracy of the obtained estimators to illustrate the usefulness of the proposed methodology. To illustrate the usefulness of the proposed model, we also include an example with real data from which it is possible to see that the proposed model leads to good fits to the data.


Achcar, J. A. and Leandro, R. A. (1998). Use of Markov Chain Monte Carlo methods in a Bayesian analysis of the Block and Basu bivariate exponential distribution. Annals of the Institute of Statistical Mathematics, 50(3):403–416.

Achcar, J. A., Martinez, E. Z., and Tovar Cuevas, J. R. (2016). Bivariate lifetime modelling using copula functions in presence of mixture and non-mixture cure fraction models, censored data and covariates. Model Assisted Statistics and Applications, 11(4):261–276.

Al Kadiri, M. and Migdadi, M. (2019). Estimating parameters of Morgenstern type bivariate distribution using bivariate ranked set sampling. Electronic Journal of Applied Statistical Analysis, 12(1):190–208.

Arnold, B. C. and Strauss, D. (1988). Bivariate distributions with exponential conditionals. Journal of the American Statistical Association, 83(402):522–527.

Balakrishnan, N. (2014). Continuous multivariate distributions. Wiley StatsRef: Statistics Reference Online.

Balakrishnan, N. and Lai, C. (2009). Continuous Bivariate Distributions. Springer, New York.

Balakrishnan, N. and Risti´ c, M. M. (2016). Multivariate families of gamma-generated distributions with finite or infinite support above or below the diagonal. Journal of Multivariate Analysis, 143:194–207.

Basu, A. (1971). Bivariate failure rate. Journal of the American Statistical Association, 66(333):103–104.

Block, H. W. and Basu, A. (1974). A continuous, bivariate exponential extension. Journal of the American Statistical Association, 69(348):1031–1037.

Brown, W. K. and Wohletz, K. H. (1995). Derivation of the Weibull distribution based on physical principles and its connection to the Rosin–Rammler and lognormal distributions. Journal of Applied Physics, 78(4):2758– 2763.

Cao, Q. V. (2004). Predicting parameters of a Weibull function for modeling diameter distribution. Forest science, 50(5):682–685.

Carlin, B. P. and Louis, T. A. (2010). Bayes and empirical Bayes methods for data analysis. Chapman and Hall/CRC.

Chacko, M. and Thomas, P. Y. (2006). Concomitants of record values arising from Morgenstern type bivariate logistic distribution and some of their applications in parameter estimation. Metrika, 64(3):317–331.

Chacko, M. and Thomas, P. Y. (2009). Estimation of parameters of Morgenstern type bivariate logistic distribution by ranked set sampling. Journal of the Indian Society of Agricultural Statistics, 63(1):77–83.

Cohen, A. C. (1965). Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples. Technometrics, 7(4):579–588.

Crowder, M. J. (2012). Multivariate survival analysis and competing risks. Chapman and Hall/CRC.

de Oliveira, R. P. and Achcar, J. A. (2018). Basu-Dhar’s bivariate geometric distribution in presence of censored data and covariates: some computational aspects. Electronic Journal of Applied Statistical Analysis, 11(1):108–136.

de Oliveira, R. P., Achcar, J. A., Peralta, D., and Mazucheli, J. (2019). Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach. Journal of Applied Statistics, 46(3):449–467.

de Oliveira, R. P., Peres, M. V. d. O., Achcar, J. A., and Davarzani, N. (2020). Inference for the trivariate Marshall-Olkin Weibull distribution in presence of right-censored data. Chilean Journal of Statistics, 11(2):95–116.

de Oliveira Peres, M. V., Achcar, J. A., and Martinez, E. Z. (2020). Bivariate lifetime models in presence of cure fraction: a comparative study with many different copula functions. Heliyon, 6(6):e03961.

D’este, G. (1981). A Morgenstern-type bivariate gamma distribution. Biometrika, 68(1):339–340.

dos Santos, C. A. and Achcar, J. A. (2011). A bayesian analysis for the Block and Basu bivariate exponential distribution in the presence of covariates and censored data. Journal of Applied Statistics, 38(10):2213–2223.

Emura, T., Matsui, S., and Rondeau, V. (2019). Survival Analysis with Correlated Endpoints: Joint Frailty-Copula Models. Springer.

Gupta, A. K. and Wong, C. (1984). On a Morgenstern-type bivariate gamma distribution. Metrika, 31(1):327–332.

Joe, H. (2014). Dependence Modeling with Copulas. Chapman and Hall/CRC Press.

Johnson, M. E. and Tenenbein, A. (1981). A bivariate distribution family with specified marginals. Journal of the American Statistical Association, 76(373):198–201.

Johnson, N. L. and Kotz, S. (1975). A vector multivariate hazard rate. Journal of Multivariate Analysis, 5(1):53–66.

Kundu, D. and Gupta, R. D. (2009). Bivariate generalized exponential distribution. Journal of multivariate analysis, 100(4):581–593.

Kundu, D. and Gupta, R. D. (2011). Absolute continuous bivariate generalized exponential distribution. AStA Advances in Statistical Analysis, 95(2):169–185.

Lai, C., Xie, M., and Murthy, D. (2003). A modified Weibull distribution. IEEE Transactions on reliability, 52(1):33–37.

Lai, C.-D. (2006). Constructions of discrete bivariate distributions. In Advances in Distribution Theory, Order Statistics, and Inference, pages 29–58. Springer.

Lehmann, E. L. (1966). Some concepts of dependence. The Annals of Mathematical Statistics, 37(5):1137– 1153.

Mahmoudi, E. and Mahmoodian, H. (2017). A new bivariate distribution obtained by compounding the bivariate normal and geometric distributions. Journal of Statistical Theory and Applications, 16(2):198–208.

Marshall, A. W. and Olkin, I. (1967). A generalized bivariate exponential distribution. Journal of Applied Probability, 4(2):291–302.

Martinez, E. Z. and Achcar, J. A. (2014). Bayesian bivariate generalized lindley model for survival data with a cure fraction. Computer methods and programs in biomedicine, 117(2):145–157.

Meintanis, S. G. (2007). Test of fit for Marshall–Olkin distributions with applications. Journal of Statistical Planning and Inference, 137(12):3954–3963.

Morgenstern, D. (1956). Einfache beispiele zweidimensionaler verteilungen. Mitteilingsblatt fur Mathematische Statistik, 8:234–235.

Mudholkar, G. S., Srivastava, D. K., and Kollia, G. D. (1996). A generalization of the Weibull distribution with application to the analysis of survival data. Journal of the American Statistical Association, 91(436):1575–1583.

Nair, N. U., Sankaran, P., and John, P. (2018). Modelling bivariate lifetime data using copula. Metron, 76(2):133–153.

Nelsen, R. B. (1999). An Introduction to Copulas. Springer.

Oliveira, R. P., Achcar, J. A., Mazucheli, J., and Bertoli, W. (2021). A new class of bivariate lindley distributions based on stress and shock models and some of their reliability properties. Reliability Engineering & System Safety, 211:107528.

Ota, S. and Kimura, M. (2021). Effective estimation algorithm for parameters of multivariate farlie–gumbel– morgenstern copula. Japanese Journal of Statistics and Data Science. doi:10.1007/s42081-021-00118-y.

Peres, M. V. d. O., Achcar, J. A., and Martinez, E. Z. (2018). Bivariate modified Weibull distribution derived from Farlie Gumbel-Morgenstern copula: a simulation study. Electronic Journal of Applied Statistical Analysis, 11(2):463–488.

Pham, H. and Lai, C.-D. (2007). On recent generalizations of the Weibull distribution. IEEE transactions on reliability, 56(3):454–458.

Philip, G. C. (1974). A generalized EOQ model for items with Weibull distribution deterioration. AIIE transactions, 6(2):159–162.

Pinder III, J. E., Wiener, J. G., and Smith, M. H. (1978). The Weibull distribution: a new method of summarizing survivorship data. Ecology, 59(1):175–179.

Portilla Yela, J. and Tovar Cuevas, J. R. (2018). Estimating the Gumbel-Barnett copula parameter of dependence. Revista Colombiana de Estad´ıstica, 41(1):53–73.

Qiao, H. and Tsokos, C. P. (1995). Estimation of the three parameter Weibull probability distribution. Mathematics and Computers in Simulation, 39(1-2):173–185.

Rinne, H. (2008). The Weibull distribution: a handbook. Chapman and Hall/CRC.

Risti´ c, M. M., Popovi´ c, B. V., Zografos, K., and Balakrishnan, N. (2018). Discrimination among bivariate beta- generated distributions. Statistics, 52(2):303–320.

Rockette, H., Antle, C., and Klimko, L. A. (1974). Maximum likelihood estimation with the Weibull model. Journal of the American Statistical Association, 69(345):246–249.

Romeo, J. S., Meyer, R., and Gallardo, D. I. (2018). Bayesian bivariate survival analysis using the power variance function copula. Lifetime data analysis, 24(2):355–383.

Roy, D. (2004). Bivariate models from univariate life distributions: a characterization cum modeling approach. Naval Research Logistics (NRL), 51(5):741–754.

Scaria, J. and Nair, N. U. (1999). On concomitants of order statistics from Morgenstern family. Biometrical Journal: Journal of Mathematical Methods in Biosciences, 41(4):483–489.

Shih, J.-H., Konno, Y., Chang, Y.-T., and Emura, T. (2019). Estimation of a common mean vector in bivariate meta analysis under the fgm copula. Statistics, 53(3):673–695.

Sklar, M. (1959). Fonctions de repartition an dimensions et leurs marges. Publ. inst. statist. univ. Paris, 8:229–231.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4):583–639.

Stevens, M. and Smulders, P. (1979). The estimation of the parameters of the Weibull wind speed distribution for wind energy utilization purposes. Wind Engineering, 3(2):132–145.

Su, Y.-S. and Yajima, M. (2012). R2jags: A package for running jags from r. R package version 0.03-08, URL http://CRAN. R-project. org/package= R2jags.

Tahmasebi, S. and Jafari, A. A. (2012). Estimation of a scale parameter of Morgenstern type bivariate uniform distribution by ranked set sampling. Journal of Data Science, 10(1):129–141.

Teimouri, M. and Gupta, A. K. (2013). On the three-parameter Weibull distribution shape parameter estimation. Journal of Data Science, 11(3):403–414.

Thoman, D. R., Bain, L. J., and Antle, C. E. (1969). Inferences on the parameters of the Weibull distribution. Technometrics, 11(3):445–460.

Vaidyanathan, V., Sharon Varghese, A., et al. (2016). Morgenstern type bivariate Lindley distribution. Statistics, Optimization & Information Computing, 4(2):132–146.

Vaupel, J. W., Manton, K. G., and Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16(3):439–454.

Wang, Y.-C., Emura, T., Fan, T.-H., Lo, S. M., and Wilke, R. A. (2020). Likelihood-based inference for a frailty-copula model based on competing risks failure time data. Quality and Reliability Engineering International, 36(5):1622–1638.

Weibull, W. (1951). Wide applicability. Journal of Applied Mechanics, 103(730):293–297.

Zanakis, S. H. (1979). A simulation study of some simple estimators for the three-parameter Weibull distribution. Journal of Statistical Computation and Simulation, 9(2):101–116.

How to Cite
Ricardo Puziol de Oliveira, Marcos Vinicius de Oliveira Peres, Milene Regina dos Santos, Edson Zangiacomi Martinez, & Jorge Aberto Achcar. (2021). A Bayesian Inference Approach for Bivariate Weibull Distributions Derived from Roy and Morgenstern Methods. Statistics, Optimization & Information Computing, 9(3), 529-554. https://doi.org/10.19139/soic-2310-5070-1240
Research Articles