# On a Bivariate XGamma Distribution Derived from Copula

## DOI:

https://doi.org/10.6092/issn.1973-2201/12293## Keywords:

Bivariate XGamma distribution, Copulas, FGM copula, Maximum likelihood estimate, Inference function of margin## Abstract

In this paper, a new bivariate XGamma (BXG) distribution is presented using Farlie-Gumbel-Morgenstern (FGM) copula. We derive the expressions for conditional distribution, regression function and product moments for the BXG distribution. Concept of reliability and various measures of local dependence are also studied for the proposed model. Furthermore, estimation of the parameters of the BXG distribution is obtained through maximum likelihood estimation and inference function of margin estimation procedures. Finally, an application of the same is also demonstrated to a real data set.

## References

M. K. ABD ELAAL, R. S. JARWAN (2017). Inference of bivariate generalized Exponential distribution based on copula functions. Applied Mathematical Sciences, 11, no. 24, pp. 1155–1186.

J. A. ACHCAR, F. A.MOALA, M. H. TARUMOTO, L. F. COLADELLO (2015). A bivariate generalized Exponential distribution derived from copula functions in the presence of censored data and covariates. Pesquisa Operacional, 35, no. 1, pp. 165–186.

E. M. ALMETWALLY, H. Z. MUHAMMED, E.-S. A. EL-SHERPIENY (2020). Bivariate Weibull distribution: properties and different methods of estimation. Annals of Data Science, 7, no. 1, pp. 163–193.

C. AMBLARD, S. GIRARD (2009). A new extension of bivariate FGM copulas. Metrika, 70, no. 1, pp. 1–17.

J. E. ANDERSON, T. A. LOUIS, N. V. HOLM, B. HARVALD (1992). Time-dependent association measures for bivariate survival distributions. Journal of the American Statistical Association, 87, no. 419, pp. 641–650.

N. BALAKRISHNAN, C. D. LAI (2009). Continuous Bivariate Distributions. Springer Science & Business Media, New York.

A. BASU (1971). Bivariate failure rate. Journal of the American Statistical Association, 66, no. 333, pp. 103–104.

S. BHATTACHARJEE, S. K. MISRA (2016). Some aging properties of Weibull models. Electronic Journal of Applied Statistical Analysis, 9, no. 2, pp. 297–307.

D. G. CLAYTON (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, no. 1, pp. 141–151.

M. V. DE OLIVEIRA PERES, J. A. ACHCAR, E. Z. MARTINEZ (2020). Bivariate lifetime models in presence of cure fraction: a comparative study with many different copula functions. Heliyon, 6, no. 6, p. e03961.

A. DOLATI, M. AMINI, S. MIRHOSSEINI (2014). Dependence properties of bivariate distributions with proportional (reversed) hazards marginals. Metrika, 77, no. 3, pp. 333–347.

C. A. DOS SANTOS, J. A. ACHCAR (2010). A Bayesian analysis for bivariate Weibull distributions derived from copula functions in the presence of covariates and censored data. Advances and Applications in Statistics, 15, no. 1, pp. 1–25.

D. J. FARLIE (1960). The performance of some correlation coefficients for a general bivariate distribution. Biometrika, 47, no. 3/4, pp. 307–323.

E. J. GUMBEL (1958). Statistics of Extremes. Columbia University Press, New York.

E. J. GUMBEL (1960). Bivariate Exponential distributions. Journal of the American Statistical Association, 55, no. 292, pp. 698–707.

P. W. HOLLAND, Y. J. WANG (1987). Dependence function for continuous bivariate densities. Communications in Statistics-Theory and Methods, 16, no. 3, pp. 863–876.

H. JOE (2014). Dependence Modeling with Copulas. CRC press, New York.

H. JOE, J. J. XU (1996). The Estimation Method of Inference Functions for Margins for Multivariate Models. Technical Report No. 166, Department of Statistics, University of British Columbia, Vancouver.

N. JOHNSON, S. KOTZ, N. BALAKRISHNAN (1995). Continuous Univariate Distributions, 2nd ed. JohnWiley and Sons, New York.

N. L. JOHNSON, S. KOTZ (1975). A vector multivariate hazard rate. Journal of Multivariate Analysis, 5, no. 1, pp. 53–66.

D. KUNDU, A. K. GUPTA, et al. (2017). On bivariate inverse Weibull distribution. Brazilian Journal of Probability and Statistics, 31, no. 2, pp. 275–302.

D. KUNDU, R. C.GUPTA (2017). On bivariate Birnbaum–Saunders distribution. American Journal of Mathematical and Management Sciences, 36, no. 1, pp. 21–33.

D. KUNDU, R. D. GUPTA (2009). Bivariate generalized Exponential distribution. Journal of Multivariate Analysis, 100, no. 4, pp. 581–593.

D. KUNDU, R. D. GUPTA (2011). Absolute continuous bivariate generalized Exponential distribution. Advances in Statistical Analysis, 95, no. 2, pp. 169–185.

A. W. MARSHALL, I. OLKIN (1967). A generalized bivariate Exponential distribution. Journal of Applied Probability, 4, no. 2, pp. 291–302.

S. G. MEINTANIS (2007). Test of fit for Marshall–Olkin distributions with applications. Journal of Statistical Planning and Inference, 137, no. 12, pp. 3954–3963.

S. M. MIRHOSSEINI, M. AMINI, D. KUNDU, A. DOLATI (2015). On a new absolutely continuous bivariate generalized Exponential distribution. Statistical Methods & Applications, 24, no. 1, pp. 61–83.

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

N. U. NAIR, P. SANKARAN, P. JOHN (2018). Modelling bivariate lifetime data using copula. Metron, 76, no. 2, pp. 133–153.

H. NAJARZADEGAN, M. ALAMATSAZ, I. KAZEMI (2019). Discrete bivariate distributions generated by copulas: Dbeew distribution. Journal of Statistical Theory and Practice, 13, no. 3, p. 47.

R. B. NELSEN (2006). An Introduction to Copulas. Springer Science & Business Media, New York.

D.OAKES (1989). Bivariate survival models induced by frailties. Journal of the American Statistical Association, 84, no. 406, pp. 487–493.

S.OTA, M. KIMURA (2021). Effective estimation algorithm for parameters of multivariate Farlie–Gumbel–Morgenstern copula. Japanese Journal of Statistics and Data Science, pp. 1–30.

A. PATHAK, P. VELLAISAMY (2016a). Various measures of dependence of a new asymmetric generalized Farlie–Gumbel–Morgenstern copulas. Communications in Statistics-Theory and Methods, 45, no. 18, pp. 5299–5317.

A. PATHAK, P. VELLAISAMY (2016b). A note on generalized Farlie-Gumbel-Morgenstern copulas. Journal of Statistical Theory and Practice, 10, no. 1, pp. 40–58.

A. PATHAK, P. VELLAISAMY (2020). A bivariate generalized linear Exponential distribution: properties and estimation. Communications in Statistics-Simulation and Computation, pp. 1–21.

M. V. D. O. PERES, J.A. ACHCAR, E. Z.MARTINEZ (2018). Bivariate modified Weibull distribution derived from Farlie-Gumbel-Morgenstern copula: a simulation study. Electronic Journal of Applied Statistical Analysis, 11, no. 2, pp. 463–488.

B. V. POPOVI´C , A. I. GENC, F. DOMMA (2018). Copula-based properties of the bivariate Dagum distribution. Computational and Applied Mathematics, 37, no. 5, pp. 6230–6251.

H. RINNE (2008). The Weibull Distribution: a Handbook. CRC press, New York.

R. G. SAMANTHI, J. SEPANSKI (2019). A bivariate extension of the Beta generated distribution derived from copulas. Communications in Statistics-Theory and Methods, 48, no. 5, pp. 1043–1059.

P. SANKARAN, N. U. NAIR (1993). A bivariate Pareto model and its applications to reliability. Naval Research Logistics, 40, no. 7, pp. 1013–1020.

E. F. SARAIVA, A. K. SUZUKI, L. A.MILAN (2018). Bayesian computational methods for sampling from the posterior distribution of a bivariate survival model, based on AMH copula in the presence of right-censored data. Entropy, 20, no. 9, p. 642.

A. M. SARHAN, D. C. HAMILTON, B. SMITH, D. KUNDU (2011). The bivariate generalized linear failure rate distribution and its multivariate extension. Computational Statistics & Data Analysis, 55, no. 1, pp. 644–654.

A. M. SARHAN, D. KUNDU (2009). Generalized linear failure rate distribution. Communications in Statistics-Theory and Methods, 38, no. 5, pp. 642–660.

S. SEN, N. CHANDRA, S. S. MAITI (2018). On properties and applications of a twoparameter XGamma distribution. Journal of Statistical Theory and Applications, 17, no. 4, pp. 674–685.

S. SEN, S. S.MAITI,N. CHANDRA (2016). The XGamma distribution: statistical properties and application. Journal of Modern Applied Statistical Methods, 15, no. 1, p. 38.

J. H. SHIH, Y. KONNO, Y.-T. CHANG, T. EMURA (2019). Estimation of a common mean vector in bivariate meta-analysis under the FGM copula. Statistics, 53, no. 3, pp. 673–695.

M. SKLAR (1959). Fonctions de repartition an dimensions et leurs marges. Publications de l’Institut Statistique de l’Université de Paris, 8, pp. 229–231.

B. TAHERI, H. JABBARI, M. AMINI (2018). Parameter estimation of bivariate distributions in presence of outliers: an application to FGM copula. Journal of Computational and Applied Mathematics, 343, pp. 155–173.

## Downloads

## Published

## How to Cite

*Statistica*,

*82*(1), 15–40. https://doi.org/10.6092/issn.1973-2201/12293

## Issue

## Section

## License

Copyright (c) 2022 Statistica

This work is licensed under a Creative Commons Attribution 3.0 Unported License.