Bivariate Modified Weibull Distribution Derived From Farlie-Gumbel-Morgenstern Copula: a Simulation Study


Abstract


In recent years, the use of copulas has grown rapidly, especially in survivalanalysis. In this paper, we introduce a bivariate modied Weibull distribu-tion derived from the Farlie{Gumbel{Morgenstern (FGM), a copula functioncommonly used to model very weak linear dependences. Considering thepresence of non censored data and censored data, an extensive simulationstudy was developed to check the performance of the maximum likelihoodmethod in estimating the parameters of the proposed model. Maximum like-lihood and Bayesian approaches for the estimation of the model parametersare presented. In the Bayesian analysis, the posterior distributions of the pa-rameters are estimated using Markov chain Monte Carlo (MCMC) method-ology. An example, considering a real data set, is introduced to illustrate theproposed methodology.

DOI Code: 10.1285/i20705948v11n2p463

Keywords: Bayesian estimates; bivariate data; copula function; simulation study; survival analysis

References


Abbas, A. E. (2006). Entropy methods for joint distributions in decision analysis. IEEE Transactions on Engineering Management, 53(1):146-159.

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,

(4):261-276.

Akaike, H. (1974). A new look at the statistical model identication. IEEE Transactions on Automatic Control, 19(6):716-723.

Ali, M. M., Mikhail, N., and Haq, M. S. (1978). A class of bivariate distributions including the bivariate logistic. Journal of Multivariate Analysis, 8(3):405-412.

Balakrishnan, N. and Lai, C.-D. (2009). Continuous bivariate Distributions. Springer Vergland.

Basu, A. P. and Dhar, S. (1995). Bivariate geometric distribution. Journal of Applied Statistical Science, 2(12):33-34.

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

Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65(1):141-151.

Corless, R. M., Gonnet, G. H., Hare, D. E., Jerey, D. J., and Knuth, D. E. (1996). On the LambertW function. Advances in Computational Mathematics, 5(1):329-359.

Farlie, D. J. (1960). The performance of some correlation coecients for a general bivariate distribution. Biometrika, 47(3/4):307-323.

Favre, A.-C., El Adlouni, S., Perreault, L., Thiemonge, N., and Bobee, B. (2004). Multivariate hydrological frequency analysis using copulas. Water Resources Research,40(1):1-12.

Goerg, G. M. (2016). LambertW: Probabilistic Models to Analyze and Gaussianize Heavy-Tailed, Skewed Data. R package version 0.6.4.

Group, D. R. S. R. et al. (1976). Preliminary report on eects of photocoagulation therapy. American Journal of Ophthalmology, 81(4):383-396.

Gumbel, E. J. (1960). Bivariate exponential distributions. Journal of the American Statistical Association, 55(292):698-707.

Henningsen, A. and Toomet, O. (2011). maxlik: A package for maximum likelihood estimation in R. Computational Statistics, 26(3):443-458.

Hougaard, P. (1987). Modelling multivariate survival. Scandinavian Journal of Statistics, 14(4):291-304.

Hougaard, P. (2012). Analysis of multivariate survival data. Springer Science & Business Media.

Jaworski, P., Durante, F., Hardle, W. K., and Rychlik, T. (2010). Copula theory and its applications. Springer.

Joe, H. (2014). Dependence modeling with copulas. CRC Press.

Lai, C., Xie, M., and Murthy, D. (2003). A modied Weibull distribution. IEEE Transactions on Reliability, 52(1):33-37.

Liang, K.-Y., Self, S. G., Bandeen-Roche, K. J., and Zeger, S. L. (1995). Some recent developments for regression analysis of multivariate failure time data. Lifetime Data

Analysis, 1(4):403-415.

Louzada, F., Suzuki, A., and Cancho, V. (2013). The FGM long-term bivariate survival copula model: modeling, Bayesian estimation, and case infuence diagnostics.

Communications in Statistics-Theory and Methods, 42(4):673-691.

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.

Morgenstern, D. (1956). Einfache beispiele zweidimensionaler verteilungen. Mitteilings-blatt fur Mathematische Statistik, 8:234-235.

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

Parner, E. (2001). A composite likelihood approach to multivariate survival data. Scandinavian Journal of Statistics, 28(2):295-302.

Patton, A. J. (2006). Modelling asymmetric exchange rate dependence. International Economic Review, 47(2):527-556.

Roch, O. and Alegre, A. (2006). Testing the bivariate distribution of daily equity returns using copulas. an application to the spanish stock market. Computational Statistics

& Data Analysis, 51(2):1312-1329.

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

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.

Viswanathan, B. and Manatunga, A. K. (2001). Diagnostic plots for assessing the frailty distribution in multivariate survival data. Lifetime Data Analysis, 7(2):143-155.

Zhang, L. and Singh, V. (2006). Bivariate flood frequency analysis using the copula method. Journal of Hydrologic Engineering, 11(2):150-164.


Full Text: pdf
کاغذ a4
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.