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A new wider family of continuous models: The Extended Cordeiro and de Castro Family

Year 2018, Volume: 47 Issue: 4, 937 - 961, 01.08.2018

Abstract

We introduce and study general mathematical properties of a new generator of continuous distributions with three extra parameters called the extended Cordeiro and de Castro family. We investigate the asymptotes and shapes. The new density function can be expressed as a linear
combination of exponentiated densities based on the same underlying distribution. We derive a power series for the quantile function of this
family. We determine explicit expressions for the ordinary and incomplete moments, quantile and generating functions, asymptotic distribution
of the extreme values, Shannon and Rényi entropies and order statistics, which hold for any baseline model. We discuss the estimation of the model parameters by maximum likelihood and illustrate the potentiality of the introduced family by means of two applications to real data.

References

  • Alzaatreh, A., Lee C. and Famoye, F. A new method for generating families of distributions, Metron 71, 63-79, 2013.
  • Bourguignon, M. Silva, R.B., Zea, L.M. and Cordeiro, G.M. The Kumaraswamy Pareto distribution, J Stat Theor Appl 12, 1-21, 2013.
  • Chen, G. and Balakrishnan, N. A general purpose approximate goodness-of-fit test, Journal of Quality Technology 27, 154-161, 1995.
  • Cordeiro, G.M. and de Castro, M. A new family of generalized distributions, J Statist Comput Simulation 81, 883-898, 2011.
  • Cordeiro, G.M., Gomes, A.E., da-Silva, C.Q. and Ortega, E.M.M. The beta exponentiated Weibull distribution, Journal of Statistical Computation and Simulation 83, 114-138, 2013.
  • Cordeiro, G.M., Ortega, E.M. and Silva, G.O. The Kumaraswamy modified Weibull dis- tribution: theory and applications, Journal of Statistical Computation and Simulation 84, 1387-1411, 2014.
  • Cox D.R. and Hinkley D.V. Theoretical Statistics, Chapman and Hall, London, 1974.
  • Doornik J. Ox 5: object-oriented matrix programming language, fth ed. Timberlake Consultants, London, 2007.
  • Eugene, N., Lee, C. and Famoye F. Beta-normal distribution and its applications, Commun Stat Theor Methods 31, 497-512, 2002.
  • Gomes, A.E., da-SILVA, C.Q., Cordeiro, G.M. and Ortega, E.M.M. A new lifetime model: the Kumaraswamy generalized Rayleigh distribution, J Statist Comput Simulation 84, 290- 309, 2012.
  • Gradshteyn, I.S. and Ryzhik, I.M. Table of integrals, series, and products, Academic Press, San Diego, 2000.
  • Gupta, R.C. and Gupta R.D. Proportional reversed hazard rate model and its applications, J Statist Plann Inference 137, 3525-3536, 2007.
  • Gupta R.D. and Kundu, D. Exponentiated exponential family: an alternative to gamma and Weibull, Biometrical J 43, 117-130, 2001.
  • Kenny, J.F. and Keeping, E.S. Skewness. Section 7.10 in Mathematics of Statistics, Pt. 1, 1962.
  • Marshall, A.W. and Olkin, I. A new method for adding a parameter to a family of distri- butions with application to the exponential and Weibull families, Biometrika 84, 641-652, 1997.
  • Moors, J.J.A. A quantile alternative for kurtosis, The statistician, 25-32, 1988.
  • Mudholkar G.S., Srivastava D.K. and Kollia, G.D. A generalization of the Weibull distribu- tion with application to the analysis of survival data, J Amer Statist Assoc 91, 1575-1583, 1996.
  • Nadarajah, S. and Kotz, S. The exponentiated type distributions, Acta Applicandae Mathematicae 92, 97-111, 2006.
  • Nadarajah, S. and Kotz, S. On the alternative to the Weibull function, Eng Fracture Mech 74, 451-456, 2007.
  • Paranaíba P.F., Ortega, E.M.M., Cordeiro, G.M. and Pascoa, M.A.R. The Kumaraswamy Burr XII distribution: theory and practice, J Statist Comput Simulation 82, 1-27, 2012.
  • Pascia, M.A.R., Ortega, E.M.M. and Cordeiro, G.M. The Kumaraswamy generalized gamma distribution with application in survival analysis, Stat Methodol 8, 411-433, 2011.
  • Prudnikov, A.P., Brychkov, Y.A. and Marrichev, O.I. Integrals and series, 1986. Gordon and Breach Science Publishers, Amsterdam, 1986.
  • Rényi A. On measures of information and entropy, Proc 4th Berkeley Symp Math Stat Prob 547-561, 1961.
  • Shannon, C.E. A mathematical theory of communication, Bell Syst Tech J 27, 379-432, 1948.
  • Silva, R.B., Barreto-Souza W. and Cordeiro, G.M. A new distribution with decreasing, increasing and upside-down bathtub failure rate, Comp Stat Data Anal 54, 935-944, 2010.
  • Smith R.L. and Naylor, J. A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Appl Stat 36, 358-369, 1987.
  • Wright, E.M. The asymptotic expansion of the generalized hypergeometric function, Proc London Math Soc 10, 286-293, 1935.
  • Zografos, K. and Balakrishnan N. On families of beta and generalized gamma generated distributions and associated inference, Stat Methodol 6, 344-362, 2009.
Year 2018, Volume: 47 Issue: 4, 937 - 961, 01.08.2018

Abstract

References

  • Alzaatreh, A., Lee C. and Famoye, F. A new method for generating families of distributions, Metron 71, 63-79, 2013.
  • Bourguignon, M. Silva, R.B., Zea, L.M. and Cordeiro, G.M. The Kumaraswamy Pareto distribution, J Stat Theor Appl 12, 1-21, 2013.
  • Chen, G. and Balakrishnan, N. A general purpose approximate goodness-of-fit test, Journal of Quality Technology 27, 154-161, 1995.
  • Cordeiro, G.M. and de Castro, M. A new family of generalized distributions, J Statist Comput Simulation 81, 883-898, 2011.
  • Cordeiro, G.M., Gomes, A.E., da-Silva, C.Q. and Ortega, E.M.M. The beta exponentiated Weibull distribution, Journal of Statistical Computation and Simulation 83, 114-138, 2013.
  • Cordeiro, G.M., Ortega, E.M. and Silva, G.O. The Kumaraswamy modified Weibull dis- tribution: theory and applications, Journal of Statistical Computation and Simulation 84, 1387-1411, 2014.
  • Cox D.R. and Hinkley D.V. Theoretical Statistics, Chapman and Hall, London, 1974.
  • Doornik J. Ox 5: object-oriented matrix programming language, fth ed. Timberlake Consultants, London, 2007.
  • Eugene, N., Lee, C. and Famoye F. Beta-normal distribution and its applications, Commun Stat Theor Methods 31, 497-512, 2002.
  • Gomes, A.E., da-SILVA, C.Q., Cordeiro, G.M. and Ortega, E.M.M. A new lifetime model: the Kumaraswamy generalized Rayleigh distribution, J Statist Comput Simulation 84, 290- 309, 2012.
  • Gradshteyn, I.S. and Ryzhik, I.M. Table of integrals, series, and products, Academic Press, San Diego, 2000.
  • Gupta, R.C. and Gupta R.D. Proportional reversed hazard rate model and its applications, J Statist Plann Inference 137, 3525-3536, 2007.
  • Gupta R.D. and Kundu, D. Exponentiated exponential family: an alternative to gamma and Weibull, Biometrical J 43, 117-130, 2001.
  • Kenny, J.F. and Keeping, E.S. Skewness. Section 7.10 in Mathematics of Statistics, Pt. 1, 1962.
  • Marshall, A.W. and Olkin, I. A new method for adding a parameter to a family of distri- butions with application to the exponential and Weibull families, Biometrika 84, 641-652, 1997.
  • Moors, J.J.A. A quantile alternative for kurtosis, The statistician, 25-32, 1988.
  • Mudholkar G.S., Srivastava D.K. and Kollia, G.D. A generalization of the Weibull distribu- tion with application to the analysis of survival data, J Amer Statist Assoc 91, 1575-1583, 1996.
  • Nadarajah, S. and Kotz, S. The exponentiated type distributions, Acta Applicandae Mathematicae 92, 97-111, 2006.
  • Nadarajah, S. and Kotz, S. On the alternative to the Weibull function, Eng Fracture Mech 74, 451-456, 2007.
  • Paranaíba P.F., Ortega, E.M.M., Cordeiro, G.M. and Pascoa, M.A.R. The Kumaraswamy Burr XII distribution: theory and practice, J Statist Comput Simulation 82, 1-27, 2012.
  • Pascia, M.A.R., Ortega, E.M.M. and Cordeiro, G.M. The Kumaraswamy generalized gamma distribution with application in survival analysis, Stat Methodol 8, 411-433, 2011.
  • Prudnikov, A.P., Brychkov, Y.A. and Marrichev, O.I. Integrals and series, 1986. Gordon and Breach Science Publishers, Amsterdam, 1986.
  • Rényi A. On measures of information and entropy, Proc 4th Berkeley Symp Math Stat Prob 547-561, 1961.
  • Shannon, C.E. A mathematical theory of communication, Bell Syst Tech J 27, 379-432, 1948.
  • Silva, R.B., Barreto-Souza W. and Cordeiro, G.M. A new distribution with decreasing, increasing and upside-down bathtub failure rate, Comp Stat Data Anal 54, 935-944, 2010.
  • Smith R.L. and Naylor, J. A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Appl Stat 36, 358-369, 1987.
  • Wright, E.M. The asymptotic expansion of the generalized hypergeometric function, Proc London Math Soc 10, 286-293, 1935.
  • Zografos, K. and Balakrishnan N. On families of beta and generalized gamma generated distributions and associated inference, Stat Methodol 6, 344-362, 2009.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Statistics
Authors

Gauss M. Cordeiro

Morad Alizadeh

Rodrigo B. Silva This is me

Thiago G. Ramires

Publication Date August 1, 2018
Published in Issue Year 2018 Volume: 47 Issue: 4

Cite

APA Cordeiro, G. M., Alizadeh, M., Silva, R. B., Ramires, T. G. (2018). A new wider family of continuous models: The Extended Cordeiro and de Castro Family. Hacettepe Journal of Mathematics and Statistics, 47(4), 937-961.
AMA Cordeiro GM, Alizadeh M, Silva RB, Ramires TG. A new wider family of continuous models: The Extended Cordeiro and de Castro Family. Hacettepe Journal of Mathematics and Statistics. August 2018;47(4):937-961.
Chicago Cordeiro, Gauss M., Morad Alizadeh, Rodrigo B. Silva, and Thiago G. Ramires. “A New Wider Family of Continuous Models: The Extended Cordeiro and De Castro Family”. Hacettepe Journal of Mathematics and Statistics 47, no. 4 (August 2018): 937-61.
EndNote Cordeiro GM, Alizadeh M, Silva RB, Ramires TG (August 1, 2018) A new wider family of continuous models: The Extended Cordeiro and de Castro Family. Hacettepe Journal of Mathematics and Statistics 47 4 937–961.
IEEE G. M. Cordeiro, M. Alizadeh, R. B. Silva, and T. G. Ramires, “A new wider family of continuous models: The Extended Cordeiro and de Castro Family”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 4, pp. 937–961, 2018.
ISNAD Cordeiro, Gauss M. et al. “A New Wider Family of Continuous Models: The Extended Cordeiro and De Castro Family”. Hacettepe Journal of Mathematics and Statistics 47/4 (August 2018), 937-961.
JAMA Cordeiro GM, Alizadeh M, Silva RB, Ramires TG. A new wider family of continuous models: The Extended Cordeiro and de Castro Family. Hacettepe Journal of Mathematics and Statistics. 2018;47:937–961.
MLA Cordeiro, Gauss M. et al. “A New Wider Family of Continuous Models: The Extended Cordeiro and De Castro Family”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 4, 2018, pp. 937-61.
Vancouver Cordeiro GM, Alizadeh M, Silva RB, Ramires TG. A new wider family of continuous models: The Extended Cordeiro and de Castro Family. Hacettepe Journal of Mathematics and Statistics. 2018;47(4):937-61.