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Families of Distributions Arising from Distributions of Ordered Data

  • Ahmadi, Mosayeb (Department of Mathematics and Computer Sciences, Damghan University) ;
  • Razmkhah, M. (Department of Statistics, Ferdowsi University of Mashhad) ;
  • Mohtashami Borzadaran, G.R. (Department of Statistics, Ferdowsi University of Mashhad)
  • Received : 2014.02.13
  • Accepted : 2015.01.12
  • Published : 2015.03.31

Abstract

A large family of distributions arising from distributions of ordered data is proposed which contains other models studied in the literature. This extension subsume many cases of weighted random variables such as order statistics, records, k-records and many others in variety. Such a distribution can be used for modeling data which are not identical in distribution. Some properties of the theoretical model such as moment, mean deviation, entropy criteria, symmetry and unimodality are derived. The proposed model also studies the problem of parameter estimation and derives maximum likelihood estimators in a weighted gamma distribution. Finally, it will be shown that the proposed model is the best among the previously introduced distributions for modeling a real data set.

References

  1. Akinsete, A., Famoye, F. and Lee, C. (2008). The beta-Pareto distribution, Statistics, 42, 547-563. https://doi.org/10.1080/02331880801983876
  2. Amini, M., MirMostafaee, S. M. T. K. and Ahmadi, J. (2014). Log-Gamma generated families of distributions, Statistics, 48, 913-932. https://doi.org/10.1080/02331888.2012.748775
  3. Amusan, G. E. (2010). The Beta Maxwell Distribution, Marshall University, Thesis.
  4. Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (2008). A First Course in Order Statistics, SIAM, Philadelphia.
  5. Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records, JohnWiley & Sons, New York.
  6. Bartoszewicz, J. (2009). On a representation of weighted distributions, Statistics and probability Letters, 79, 1690-1694. https://doi.org/10.1016/j.spl.2009.04.007
  7. Barreto-Souza, W., Santos, A. and Cordeiro, G. M. (2010). The beta generalized exponential distri-bution, Journal of Statistical Computation and Simulation, 80, 159-172. https://doi.org/10.1080/00949650802552402
  8. Castellares, F., Montenegro, L. C. and Cordeiro, G. M. (2013). The beta log-normal distribution, Journal of Statistical Computation and Simulation, 83, 203-228. https://doi.org/10.1080/00949655.2011.599809
  9. Cordeiro, G. M. and Lemonte, A. J. (2011). The beta Laplace distribution, Statistics and Probability Letters, 81, 973-82. https://doi.org/10.1016/j.spl.2011.01.017
  10. Cordeiro, G. M., Ortega, E. M. M. and Silva, G. O. (2012). The beta extendedWeibull family, Journal of Probability and Statistical Science, 10, 15-40.
  11. David, H. A. and Nagaraja, H. N. (2003). Order Statistics, Third Edition, Wiley, New Jersey,
  12. Fischer, M. J. and Vaughan, D. (2010). The beta-hyperbolic secant distribution, Austrian Journal of Statistics, 39, 245-258.
  13. Fisher, R. A. (1934). The effect of methods of ascertainment upon the estimation of frequencies. Annals of Eugenics, 6, 13-25. https://doi.org/10.1111/j.1469-1809.1934.tb02105.x
  14. Gusmao, F. R. S., Ortega, E. M. M. and Cordeiro, G. M. (2011). The generalized inverse Weibull distribution, Statistical Papers, 52, 591-619. https://doi.org/10.1007/s00362-009-0271-3
  15. Jones, M. C. (2004). Families of distributions arising from distributions of order statistics (with discussion), Test, 13, 1-43. https://doi.org/10.1007/BF02602999
  16. Lemonte, A. J. (2014). The beta log-logistic distribution, Brazilian Journal of Probability and Statistics, 28, 313-332. https://doi.org/10.1214/12-BJPS209
  17. Moraisa, A. L., Cordeiro, G. M. and Cysneiros, A. H. M. A. (2013). The Beta generalized logistic distribution, Brazilian Journal of Probability and Statistics, 27, 185-200. https://doi.org/10.1214/11-BJPS166
  18. Nadarajah, S., Bakouch, H. S. and Tahmasbi, R. (2011). A generalized Lindley distribution, Sankhya B, 73, 331-359. https://doi.org/10.1007/s13571-011-0025-9
  19. Nadarajah, S. and Kotz, S. (2004). The beta Gumbel distribution, Mathematical Problems in Engineering, 10, 323-332.
  20. Nadarajah, S. and Gupta, A. K. (2004). The beta Frechet distribution, Far East Journal of Theoretical Statistics, 14, 15-24.
  21. Parnaiba, P. F., Ortega, E. M. M., Cordeiro, G. M. and Pescim, R. R. (2011). The beta Burr XII distribution with application to lifetime data, Computational Statistics and Data Analysis, 55, 1118-1136. https://doi.org/10.1016/j.csda.2010.09.009
  22. Pescim, R. R., Demerio, C. G. B., Cordeiro, G. M., Ortega, E. M. M. and Urbanoa, M. R. (2010). The beta generalized half-normal distribution, Computational Statistics and Data Analysis, 54, 945-957. https://doi.org/10.1016/j.csda.2009.10.007
  23. Ramadan, M. M. (2013). A class of weighted Weibull distributions and its properties, Studies in Mathematical Sciences, 6, 35-45.
  24. Reiss, R. D. (1989). Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics, Springer-Verlag, Berlin.
  25. Rao, C. R. (1965). Weighted distributions arising out of methods of ascertainment. In Classical and Contagious Discrete Distributions, G. P. Patil (Eds). Calcutta: Pergamon Press and Statistical Publishing Society, 320-332.
  26. Shannon, C. E. (1948). A mathematical theory of communication, Bell System Technical Journal, 27, 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
  27. Silva, G. O., Ortega, E. M. M. and Cordeiro, G. M. (2010). The beta modified Weibull, Distribution, Lifetime Data Analysis, 16, 409-430. https://doi.org/10.1007/s10985-010-9161-1
  28. Ye, Y., Oluyede, B. O. and Pararai, M. (2012). Weighted generalized beta distribution of the second kind and related distributions, Journal of Statistical and Econometric Methods, 1, 13-31.

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