The Exponentiated Weibull-Geometric Distribution: Properties and Estimations

- Journal title : Communications for Statistical Applications and Methods
- Volume 21, Issue 2, 2014, pp.147-160
- Publisher : The Korean Statistical Society
- DOI : 10.5351/CSAM.2014.21.2.147

Title & Authors

The Exponentiated Weibull-Geometric Distribution: Properties and Estimations

Chung, Younshik; Kang, Yongbeen;

Chung, Younshik; Kang, Yongbeen;

Abstract

In this paper, we introduce the exponentiated Weibull-geometric (EWG) distribution which generalizes two-parameter exponentiated Weibull (EW) distribution introduced by Mudholkar et al. (1995). This proposed distribution is obtained by compounding the exponentiated Weibull with geometric distribution. We derive its cumulative distribution function (CDF), hazard function and the density of the order statistics and calculate expressions for its moments and the moments of the order statistics. The hazard function of the EWG distribution can be decreasing, increasing or bathtub-shaped among others. Also, we give expressions for the Renyi and Shannon entropies. The maximum likelihood estimation is obtained by using EM-algorithm (Dempster et al., 1977; McLachlan and Krishnan, 1997). We can obtain the Bayesian estimation by using Gibbs sampler with Metropolis-Hastings algorithm. Also, we give application with real data set to show the flexibility of the EWG distribution. Finally, summary and discussion are mentioned.

Keywords

Bayesian estimation;EM Algorithm;exponentiated Weibull distribution;exponentiated Weibull geometric distribution;geometric distribution;Gibbs sampler;hazard function;Metropolis-Hastings algorithm;MLE;Markov chain Monte Carlo (MCMC);

Language

English

References

1.

Alexander, C., Cordeiro, G. M., Ortega, E. M. M. and Sarabia, J. M. (2012). Generalized beta-generated distributions, Computational Statistics and Data Analysis, 56, 1880-1897.

2.

Azzalini, A. and Capitanio, A. (2003). Distribution generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution, Journal of the Royal Statistical Society, Series B, 65, 367-389.

3.

Barreto-Souza, W., Lemos de Morais, A. and Cordeiro, G. M. (2011). The Weibull geometric distribution, Journal of Statistical Computation and Simulation, 81, 645-657.

4.

Bidram, H., Behboodian, J. and Towhidi, M. (2011) The beta Weibull-geometric distribution, Jornal of Statistical Computation and Simulation, 83, 52-67.

5.

Cordeiro, G. M., Silva, G. O. and Ortega, M. M. (2011) The beta-Weibull geometric distribution, Statistics, DOI:10.1080/02331888.2011.577897

6.

Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society, Series B (Methodological), 39, 1-38.

7.

Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications, Communications in Statistics - Theory and Methods, 31, 497-512.

8.

Fonseca, M. B. and Franca, M. G. C. (2007). A in uencia da fertilidade do solo e caracterizacao da xacao biologica de N2 para o crescimento de Dimorphandra wilsonii rizz, Master's thesis, Universidade Federal de Minas Gerais.

9.

Gelfand, E. and Smith, F. M. (1990). Sampling-Based approaches to calculating marginal densities, Journal of the American Statistical Association, 85, 398-409.

10.

Hansen, B. E. (1994). Autoregressive conditional density estimation, International Economic Review, 35, 705-730.

11.

Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 97-109.

12.

Jung, J. and Chung, Y. (2013). Bayesian prediction of Exponential Weibull distribution based on progressive type II cednsoring, Communications for Statistical Applications and Methods, 20, 427-438.

13.

Kim, C., Jung, J. and Chung, Y. (2011). Bayesian estimation for the exponentiated Weibull model under Type II progressive censoring, Statistical Papers, 52, 53-70.

14.

Kwan, K. C., Breault, G. O., Umbenhauer, E. R., McMahon, F. G. and Duggan, D. F. (1976). Kinetics of Indomethacin absorption, elimination, and enterohepatic circulation in man, Journal of Pharmacokinetics and Biopharmaceutics, 4, 255-280.

15.

Mahmoudi, E. and Shiran, M. (2012). Exponentiated Weibull-Geometric Distribution and its Applications, arXiv:1206.4008vl [stat.ME]

16.

McLachlan, G. J. and Krishnan, T. (1997). The EM Algorithm and Extension, Wiley, New York.

17.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines, Journal of Chemical Physics, 21, 1087-1092.

18.

Mudholkar, G. S., Srivastava, D. K. and Freimer, M. (1995). The exponentiated Weibull family; a reanalysis of the bus motor failure data, Technometrics, 37, 436-445.

19.

Nadarajah, S., Cordeiro, G. M. and Ortega, E. M. M. (2013). The exponentiated Weibull distribution:A survey, Statistical Papers, 54, 839-877.

20.

Nassar, M. M. and Eissa, F. H. (2003). On the exponentiated Weibull distribution, Communications in Statistics - Theory and Methods, 32, 1317-1336. doi:10.1081/STA-120021561.

21.

Nassar, M. M. and Eissa, F. H. (2004). Bayesian estimation for the exponentiated Weibull model, Communications in Statistics - Theory and Methods, 33, 2343-2362.