Advertisement

Journal of Statistical Theory and Practice

, Volume 11, Issue 1, pp 179–207 | Cite as

The Gompertz-G family of distributions

  • Morad Alizadeh
  • Gauss M. Cordeiro
  • Luis Gustavo Bastos Pinho
  • Indranil GhoshEmail author
Article

Abstract

We introduce and study some general mathematical properties of a new generator of continuous distributions with two extra parameters called the Gompertz-G generator. We present some special models. We investigate the shapes of the density and hazard functions and derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, probability weighted moments, Bonferroni and Lorenz curves, Shannon and Rényi entropies, and order statistics. Two bivariate extensions of this model are proposed. We discuss the estimation of the model parameters by maximum likelihood and prove empirically the potentiality of the new class by means of two real data sets.

Keywords

Generated family Gompertz distribution maximum likelihood moment order statistic probability weighted moment quantile function Rényi entropy 

AMS Subject Classification

62E 60F 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aarts, R. M. 2000. Lauricella functions. From Math World, A Wolfram Web Resource, created by Eric W. Weisstein. https://doi.org/mathworld.wolfram.com/LauricellaFunctions.html.
  2. Alexander, C., G. M. Cordeiro, Ε. Μ. Ortega, and J. M. Sarabia. 2012. Generalized betagenerated distributions. Computational Statistics & Data Analysis 56 (6):1880–1897.MathSciNetCrossRefGoogle Scholar
  3. Alizadeh, M., M. Emadi, M. Doostparast, G. M. Cordeiro, Ε. Μ. Ortega, and R. R. Pescim. 2015. A new family of distributions: the Kumaraswamy odd log-logistic, properties and applications. Hacettepa Journal of Mathematics and Statistics.Google Scholar
  4. Alizadeh, M., M. H. Tahir, G. M. Cordeiro, Μ. Mansoor, Μ. Zubair, and G. G. Hamedani. 2015. The Kumaraswamy Marshal-Olkin family of distributions. Journal of the Egyptian Mathematical Society 23 (3):546–57.MathSciNetCrossRefGoogle Scholar
  5. Alzaatreh, Α., C. Lee, and F. Famoye. 2013. A new method for generating families of continuous distributions. Metron 71 (1):63–79.MathSciNetCrossRefGoogle Scholar
  6. Aljarrah, Μ. Α., C. Lee, and F. Famoye. 2014. On generating T-X family of distributions using quantile functions. Journal of Statistical Distributions and Applications 1 (1):1–17.CrossRefGoogle Scholar
  7. Alzaghal, Α., F. Famoye, and C. Lee. 2013. Exponentiated T-X family of distributions with some applications. International Journal of Statistics and Probability 2 (3):31.CrossRefGoogle Scholar
  8. Amini, M., S. Μ. Τ. Κ. MirMostafaee, and J. Ahmadi. 2014. Log-gamma-generated families of distributions. Statistics 48 (4):913–32.MathSciNetCrossRefGoogle Scholar
  9. Anderson, T. W., and D. A. Darling. 1954. A test of goodness of fit. Journal of the American Statistical Association 49:765–769.MathSciNetCrossRefGoogle Scholar
  10. Bourguignon, M., R. B., Silva, and G. M. Cordeiro. 2014. The Weibull-G family of probability distributions. Journal of Data Science 12:53–68.MathSciNetGoogle Scholar
  11. Cooray, K., and Μ. Μ. A. Ananda. 2008. A generalization of the half-normal distribution with applications to lifetime data. Communications in StatisticsTheory and Methods 37:1323–37.MathSciNetCrossRefGoogle Scholar
  12. Cordeiro, G. M., M. Alizadeh, and P. R. Diniz Marinho. 2016. The type I half-logistic family of distributions. Journal of Statistical Computation and Simulation 86 (4):707–28.MathSciNetCrossRefGoogle Scholar
  13. Cordeiro, G. M., M. Alizadeh, and Ε. Μ. Ortega. 2014. The exponentiated half-logistic family of distributions: Properties and applications. Journal of Probability and Statistics 2014:1–21.MathSciNetCrossRefGoogle Scholar
  14. Cordeiro, G. M., and M. de Castro. 2011. A new family of generalized distributions. Journal of Statistical Computation and Simulation 81 (7):883–98.MathSciNetCrossRefGoogle Scholar
  15. Cordeiro, G. M., and A. Lemonte. 2011. The β-Birnbaum-Saunders distribution: An improved distribution for fatigue life modeling. Computational Statistics & Data Analysis 55:1445–61.MathSciNetCrossRefGoogle Scholar
  16. Cordeiro, G. M., and S. Nadarajah. 2011. Closed-form expressions for moments of a class of beta generalized distributions. Brazilian Journal of Probability and Statistics 25:14–33.MathSciNetCrossRefGoogle Scholar
  17. Cordeiro, G. M., E. M. Ortega, and D. C. da Cunha. 2013. The exponentiated generalized class of distributions. Journal of Data Science 11 (1):1–27.MathSciNetGoogle Scholar
  18. Cordeiro, G. M., E. M. Ortega, B. V. Popovic, and R. R. Pescim. 2014. The Lomax generator of distributions: Properties, minification process and regression model. Applied Mathematics and Computation 247:465–86.MathSciNetCrossRefGoogle Scholar
  19. Cox, D. R., and D. V. Hinkley. 1974. Theoretical Statistics. London, UK: Chapman and Hall.CrossRefGoogle Scholar
  20. Doornik, J. A. 2007. An object-oriented matrix language Ox 5. London: Timberlake Consultants Press.Google Scholar
  21. Eugene, N., C. Lee, and F. Famoye. 2002. Beta-normal distribution and its applications. Communications in Statistics, Theory and Methods 31:497–512.MathSciNetCrossRefGoogle Scholar
  22. Gradshteyn, I. S., and I. M. Ryzhik. 2000. Table of integrals, series, and products. San Diego, CA: Academic Press.zbMATHGoogle Scholar
  23. Gompertz, B. 1825. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London 115:513–83.CrossRefGoogle Scholar
  24. Gupta, R. C., and R. D. Gupta. 2007 Proportional reversed hazard rate model and its applications. Journal of Statistical Planning and Inference 137:3525–3536.MathSciNetCrossRefGoogle Scholar
  25. Gupta, R. D., and R. C. Gupta. 2008. Analyzing skewed data by power normal model. Test 17 (1):197–210.MathSciNetCrossRefGoogle Scholar
  26. Gupta, R. D., and D. Kundu. 2001. Exponentiated exponential family: An alternative to gamma and Weibull. Biometrical Journal 43:117–30.MathSciNetCrossRefGoogle Scholar
  27. Jones, M. C. 2004. Families of distributions arising from distributions of order statistics. Test 13 (1):1–43.MathSciNetCrossRefGoogle Scholar
  28. Kenney, J. F., and E. S. Keeping. 1962. Mathematics of statistics, 3rd ed, part 1 101–102. Princeton, NJ: Chapman & Hall.Google Scholar
  29. Marshall, A. W., and I. Olkin. 1997. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84 (3):641–652.MathSciNetCrossRefGoogle Scholar
  30. Marshall, A. W., and I. Olkin. 2007. Life distributions. Structure of nonparametric, semiparametric and parametric families. New York: Springer.zbMATHGoogle Scholar
  31. Mudholkar, G. S., and D. K. Srivastava. 1993. Exponentiated Weibull family for analyzing bathtub failure rate data. IEEE Transactions on Reliability 42:299–302.CrossRefGoogle Scholar
  32. Mudholkar, G. S., D. K. Srivastava, and G. D. Kollia. 1996. A generalization of the Weibull distribution with application to the analysis of survival data. Journal of the American Statistical Association 91:1575–83.MathSciNetCrossRefGoogle Scholar
  33. Nadarajah, S., and S. Kotz. 2006. The exponentiated type distributions. Acta Applicandae Mathematicae 92:97–111.MathSciNetCrossRefGoogle Scholar
  34. Prudnikov, A. P., Y. A. Brychkov, and O. I. Marichev. 1986. Integrals and series. Amsterdam, The Netherlands: Gordon and Breach.zbMATHGoogle Scholar
  35. Rényi, A. 1961. On measures of information and entropy. Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 547–61.Google Scholar
  36. Ristíc, M. M., and N. Balakrishnan. 2011. The gamma-exponentiated exponential distribution. Journal Statistical Computation and Simulation 82:1191–206.MathSciNetCrossRefGoogle Scholar
  37. Ristic, M. M., and N. Balakrishnan. 2012. The gamma-exponentiated exponential distribution. Journal Statistical Computation and Simulation 82:1191–1206.MathSciNetCrossRefGoogle Scholar
  38. Shannon, C. E. 1948. A mathematical theory of communication. Bell System Technical Journal 27:379–432.MathSciNetCrossRefGoogle Scholar
  39. Tahir, M. H., G. M. Cordeiro, Μ. Alizadeh, Μ. Mansoor, Μ. Zubair, and G. G. Hamedani. 2015. The odd generalized exponential family of distributions with applications. Journal of Statistical Distributions and Applications 2 (1):1–28.CrossRefGoogle Scholar
  40. Torabi, H., and Ν. Μ. Hedesh. 2012. The gamma-uniform distribution and its applications. Kybernetika 48 (1): 16–30.MathSciNetzbMATHGoogle Scholar
  41. Torabi, H., and Ν. Η. Montazeri. 2014. The logistic-uniform distribution and its applications. Communications in Statistics—Simulation and Computation 43 (10):2551–569.MathSciNetCrossRefGoogle Scholar
  42. Zografos, K., and N. Balakrishnan. 2009. On families of beta- and generalized gamma-generated distributions and associated inference. Statistical Methodology 6 (4):344–62.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2017

Authors and Affiliations

  • Morad Alizadeh
    • 1
  • Gauss M. Cordeiro
    • 2
  • Luis Gustavo Bastos Pinho
    • 2
  • Indranil Ghosh
    • 3
    Email author
  1. 1.Department of StatisticsPersian Gulf UniversityBushehrIran
  2. 2.Universidade Federal de PernambucoPernambucoBrazil
  3. 3.Department of Mathematics and StatisticsUniversity of North CarolinaWilmingtonUSA

Personalised recommendations