Absolute continuous bivariate generalized exponential distribution
Tóm tắt
Generalized exponential distribution has been used quite effectively to model positively skewed lifetime data as an alternative to the well known Weibull or gamma distributions. In this paper we introduce an absolute continuous bivariate generalized exponential distribution by using a simple transformation from a well known bivariate exchangeable distribution. The marginal distributions of the proposed bivariate generalized exponential distributions are generalized exponential distributions. The joint probability density function and the joint cumulative distribution function can be expressed in closed forms. It is observed that the proposed bivariate distribution can be obtained using Clayton copula with generalized exponential distribution as marginals. We derive different properties of this new distribution. It is a five-parameter distribution, and the maximum likelihood estimators of the unknown parameters cannot be obtained in closed forms. We propose some alternative estimators, which can be obtained quite easily, and they can be used as initial guesses to compute the maximum likelihood estimates. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.
Tài liệu tham khảo
Bandyopadhyay, D., Basu, A.P.: On a generalization of a model by Lindley and Singpurwalla. Adv. Appl. Probab. 22, 498–500 (1990)
Basu, A.P.: Bivariate failure rate. J. Am. Stat. Assoc. 66, 103–104 (1971)
Blomqvist, N.: On a measure of dependence between two random variables. Ann. Math. Stat. 21, 593–600 (1950)
Coles, S., Hefferman, J., Tawn, J.: Dependence measures for extreme value analysis. Extremes 2, 339–365 (1999)
Domma, F.: Some properties of the bivariate Burr type III distribution. Statistics (2009, to appear). doi:10.1080/02331880902986547
Gupta, R.D., Kundu, D.: Generalized exponential distribution. Aust. N. Z. J. Stat. 41, 173–188 (1999)
Gupta, R.D., Kundu, D.: Generalized exponential distribution; existing methods and some recent developments. J. Stat. Plan. Inference 137, 3537–3547 (2007)
Gupta, R.C., Gupta, P.L., Gupta, R.D.: Modeling failure time data by Lehmann alternatives. Commun. Stat., Theory Methods 27, 887–904 (1998)
Joe, H.: Multivariate Model and Dependence Concept. Chapman & Hall, London (1997)
Johnson, N.L., Kotz, S.: A vector multivariate hazard rate. J. Multivariate Anal. 5, 53–66 (1975)
Johnson, R.A., Wichern, D.W.: Applied Multivariate Statistical Analysis, 3rd edn. Prentice Hall, New York (1992)
Kannan, N., Kundu, D., Nair, P., Tripathi, R.C.: The generalized exponential cure rate model with covariates. J. Appl. Stat. 37, 1625–1636 (2010)
Karlin, S.: Total Positivity. Stanford University Press, Stanford (1968)
Lehmann, E.L.: The power of rank test. Ann. Math. Stat. 24, 23–42 (1953)
Lindley, D.V., Singpurwalla, N.D.: Multivariate distributions for the life lengths of a system sharing a common environment. J. Appl. Probab. 23, 418–431 (1986)
Mardia, K.V.: Multivariate Pareto distributions. Ann. Math. Stat. 33, 1008–1015 (1962)
Marshall, A.W.: Some comments on hazard gradient. Stochastic Process. Appl. 3, 293–300 (1975)
Mudholkar, G.S., Srivastava, D.K., Freimer, M.: The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37, 436–445 (1995)
Nayak, T.: Multivariate Lomax distribution: Properties and usefulness in reliability theory. J. Appl. Probab. 24, 170–177 (1987)
Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)
Sankaran, P.G., Nair, N.U.: A bivariate Pareto model and its applications to reliability. Naval Res. Logist. 40, 1013–1020 (1993)
Shawky, A.I., Abu-Zinadah, H.H.: Characterizations of the exponentiated Pareto distribution based on record values. Appl. Math. Sci. 2, 1283–1290 (2008)
Surles, J.G., Padgett, W.J.: Inference for reliability and stress-strength for a scaled Burr Type X distribution. Lifetime Data Anal. 7, 187–200 (2001)