### The Odds Generalized Gamma-G Family of Distributions: Properties, Regressions and Applications

#### Abstract

In this article, a new "odds generalized gamma-G" family of distributions, called the GG-G family of distributions, is introduced. We propose a complete mathematical and statistical study of this family, with a special focus on the Fréchet distribution as baseline distribution. In particular, we provide infinite mixture representations of its probability density function and its cumulative distribution function, the expressions for the Rényi entropy, the reliability parameter and the probability density function of ith order statistic. Then, the statistical properties of the family are explored. Model parameters are estimated by the maximum likelihood method. A regression model is also investigated. A simulation study is performed to check the validity of the obtained estimators. Applications on real data sets are also included, with favorable comparisons to existing distributions in terms of goodness-of-fit.

#### Keywords

#### Full Text:

PDF (English)#### References

C. ALEXANDER, G. CORDEIRO, E. ORTEGA, J. SARABIA (2012). Generalized beta generated distributions. Computational Statistics and Data Analysis, 56, pp. 1880– 1897.

A. ALZAATREH, C. LEE, F. FAMOYE (2013). A new method for generating families of continuous distributions. Metron, 71, pp. 63–79.

A. ALZAATREH, C. LEE, F. FAMOYE (2015). Family of generalized gamma distributions: Properties and applications. Hacettepe Journal of Mathematics and Statistic, 45, pp. 869–886.

A. ALZAGHAL, C. LEE, F. FAMOYE (2013). Exponentiated T-X family of distributions with some applications. International Journal of Statistics and Probability, 2, pp. 31–49.

M. AMINI, S. MIRMOSTAFAEE, J. AHMADI (2014). Log-gamma-generated families of distributions. Statistics, 48, no. 4, pp. 913–932.

T. D. ANDRADE, L. ZEA, S. GOMES-SILVA, G. CORDEIRO (2017). The gamma generalized Pareto distribution with applications in survival analysis. International Journal of Statistics and Probability, 6, pp. 141–156.

A. ATKINSON (1985). Plots, Transformations, and Regression. Oxford University Press, Oxford.

M. BOURGUIGNON, R. SILVA, G. CORDEIRO (2014). The Weibull-G family of probability distributions. Journal of Data Science, 12, pp. 1253–1268.

C. BRITO, F. GOMES-SILVA, L. REGO, W. OLIVEIRA (2017). A new class of gamma distribution. Acta Scientiarum: Technology, 39, pp. 79–89.

R. CHANDLER, S. BATE (2007). Inference for clustered data using the independence loglikelihood. Biometrika, 94, no. 1, pp. 167–183.

C. CHESNEAU, H. BAKOUCH, T. HUSSAIN (2018). A new class of probability distributions via cosine and sine functions with applications. Communications in Statistics - Simulation and Computation, 48, no. 8, pp. 2287–2300.

R. COOK, S. WEISBERG (1982). Residuals and Influence in Regression. Chapman and Hall, New York.

G. CORDEIRO, M. ALIZADEH, E. ORTEGA (2014). The exponentiated half-logistic family of distributions: Properties and applications. Journal of Probability and Statistics, Article ID 864396, 21 pages.

G.CORDEIRO, M. DE CASTRO (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81, pp. 883–893.

G. CORDEIRO, E.ORTEGA, G. SILVA (2012). The beta extended Weibull family. Journal of Probability and Statistical Science, 10, pp. 15–40.

D.COX, E. SNELL (1968). A general definition of residuals. Journal of theRoyal Statistical Society, Series B, 30, pp. 248–265.

R. DAHIYA, J. GURLAND (1972). Goodness of fit tests for the gamma and exponential distributions. Technometrics, 14, pp. 791–801.

N. EUGENE, C. LEE, F. FAMOYE (2002). Beta-normal distribution and its applications. Commmunications in Statistics - Theory and Methods, 31, pp. 497–512.

I. GRADSHTEYN, I. RYZHIK (2000). Table of Integrals, Series and Products. Academic Press, New York.

R. GUPTA, D. KUNDU (2001). Exponentiated exponential family: An alternative to gamma and Weibull distributions. Biometrical Journal, 43, pp. 117–130.

B.HOSSEINI, M.AFSHARI, M.ALIZADEH (2018). The generalized odd gamma-G family of distributions: Properties and applications. Austrian Journal of Statistics, 47, pp. 69–89.

F. JAMAL, M. NASIR, M. TAHIR, N. MONTAZERI (2017). The odd Burr-III family of distributions. Journal of Statistics Applications and Probability, 6, pp. 105–122.

M. JONES (2004). Families of distributions arising from distributions of order statistics. Test, 13, pp. 1–43.

D. KUMAR, U. SINGH, S. SINGH (2015). A new distribution using sine function - Its application to bladder cancer patients data. Journal of Statistics Applications and Probability, 4, no. 3, pp. 417–427.

C. KUS (2007). A new lifetime distribution. Computational Statistics and Data Analysis, 51, pp. 4497–4509.

B. LANJONI, E. ORTEGA, G. CORDEIRO (2016). Extended Burr XII regression models: Theory and applications. Journal of Agricultural Biological and Environmental Statistics, 21, pp. 203–224.

J. LAWLESS (2003). Statistical Models and Methods for Lifetime Data. Wiley, New York.

C.MCCOOL (1979). Distribution of cysticercus bovis in lightly infected young cattle. Australian Veterinary Journal, 55, pp. 214–216.

M. MEAD, A. ABD-ELTAWAB (2014). A note on Kumaraswamy Fréchet distribution. Australian Journal of Basic and Applied Sciences, 8, pp. 294–300.

S. NADARAJAH, A. GUPTA (2004). The beta Fréchet distribution. Far East Journal of Theoretical Statistics, 14, pp. 15–24.

S. NADARAJAH, S. KOTZ (2003). The exponentiated Fréchet distribution. Statistics on the Internet. Http://interstat.statjournals.net/YEAR/2003/articles/0312002.pdf.

E. ORTEGA, G. PAULA, H. BOLFARINE (2008). Deviance residuals in generalized loggamma regression models with censored observations. Journal of Statistical Computation and Simulation, 78, pp. 747–764.

R. PESCIM, G. CORDEIRO, C. DEMETRIO, E. ORTEGA, S. NADARAJAH (2012). The new class of Kummer beta generalized distributions. Statistics and Operations Research Transactions (SORT), 36, pp. 153–180.

F. PRATAVIERA, E. ORTEGA, G. CORDEIRO, R. PESCIM, B. VERSSANI (2018). A new generalized odd log-logistic flexible Weibull regression model with applications in repairable systems. Reliability Engineering and System Safety, 176, pp. 13–26.

F. PROSCHAN (2000). Theoretical explanation of observed decreasing failure rate. American Statistical Society, reprint in Technometrics, 42, no. 1, pp. 7–11.

M. RISTÍC, N. BALAKRISHNAN (2012). The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82, pp. 1191–1206.

W. SHAW, I. BUCKLEY (2007). The alchemy of probability distributions: Beyond Gram-Charlier expansions and a skewkurtotic-normal distribution from a rank transmutation map. Research report, King’s College, London, U.K.

G. SILVA, E. ORTEGA, G. PAULA (2011). Residuals for log-Burr XII regression models in survival analysis. Journal of Applied and Statistics, 38, pp. 1435–1445.

H. TORABI, N. MONTAZARI (2012). The gamma-uniform distribution and its application. Kybernetika, 48, pp. 16–30.

H. TORABI, N. MONTAZARI (2014). The logistic-uniform distribution and its applications. Communications in Statistics - Simulation and Computation, 43, pp. 2551–2569.

S.WEISBERG (2005). Applied Linear Regression. Wiley, Hoboken, 3rd ed.

K. XU, M. XIE, L. TANG, S. HO (2003). Application of neural networks in forecasting engine systems reliability. Applied Soft Computing, 2, no. 4, pp. 255–268.

K. ZOGRAFOS, N. BALAKRISHNAN (2009). On families of beta- and generalized gamma generated distributions and associated inference. Statistical Methodology, 6, pp. 344–362.

DOI: 10.6092/issn.1973-2201/8665