[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.29220/CSAM.2018.25.3.245

The Marshall-Olkin generalized gamma distribution

Barriga, Gladys D.C. (Faculty of Engineering at Bauru, UNESP)
Cordeiro, Gauss M. (Department of Statistics, Federal University of Pernambuco)
Dey, Dipak K. (Department of Statistics, University of Connecticut)
Cancho, Vicente G. (Department of Applied Mathematics and Statistics, University of Sao Paulo)
Louzada, Francisco (Department of Applied Mathematics and Statistics, University of Sao Paulo)
Suzuki, Adriano K. (Department of Applied Mathematics and Statistics, University of Sao Paulo)

Publication Information

Communications for Statistical Applications and Methods / v.25, no.3, 2018 , pp. 245-261 More about this Journal

Abstract

Attempts have been made to define new classes of distributions that provide more flexibility for modelling skewed data in practice. In this work we define a new extension of the generalized gamma distribution (Stacy, The Annals of Mathematical Statistics, 33, 1187-1192, 1962) for Marshall-Olkin generalized gamma (MOGG) distribution, based on the generator pioneered by Marshall and Olkin (Biometrika, 84, 641-652, 1997). This new lifetime model is very flexible including twenty one special models. The main advantage of the new family relies on the fact that practitioners will have a quite flexible distribution to fit real data from several fields, such as engineering, hydrology and survival analysis. Further, we also define a MOGG mixture model, a modification of the MOGG distribution for analyzing lifetime data in presence of cure fraction. This proposed model can be seen as a model of competing causes, where the parameter associated with the Marshall-Olkin distribution controls the activation mechanism of the latent risks (Cooner et al., Statistical Methods in Medical Research, 15, 307-324, 2006). The asymptotic properties of the maximum likelihood estimation approach of the parameters of the model are evaluated by means of simulation studies. The proposed distribution is fitted to two real data sets, one arising from measuring the strength of fibers and the other on melanoma data.

Keywords

cure fraction model; generalized gamma distribution; geometric distribution; maximum likelihood; lifetime data;

Citations & Related Records

Reference

1	Adamidis K and Loukas S (1998). A lifetime distribution with decreasing failure rate, Statistics & Probability Letters, 39, 35-42. DOI
2	Barreto-Souza W, de Morais AL and Cordeiro GM (2011). The Weibull-Geometric distribution, Journal of Statistical Computation and Simulation, 81, 645-657. DOI
3	Berkson J and Gage RP (1952). Survival curve for cancer patients following treatment, Journal of the American Statistical Association, 47, 501-515. DOI
4	Boag JW (1949). Maximum likelihood estimates of the proportion of patients cured by cancer therapy, Journal of the Royal Statistical Society. Series B (Methodological), 11, 15-53.
5	Cooner F, Banerjee S, Carlin BP, and Sinha D (2007). Flexible cure rate modeling under latent activation schemes, Journal of the American Statistical Association, 102, 560-572. DOI
6	Cooner F, Banerjee S, and McBean AM (2006). Modelling geographically referenced survival data with a cure fraction, Statistical Methods in Medical Research, 15, 307-324. DOI
7	Cordeiro GM, Castellares F, Montenegro LC, and de Castro M (2013). The beta generalized gamma distribution, Statistics, 47, 888-900. DOI
8	Cox C, Chu H, Schneider MF, and Munoz A (2007). Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution, Statistics in Medicine, 26, 4352-4374. DOI
9	Dunn PK and Smyth GK (1996). Randomized quantile residuals, Journal of Computational and Graphical Statistics, 5, 236-244.
10	Ghitany ME (2005). Marshall-Olkin extended Pareto distribution and its application, International Journal of Applied Mathematics, 18, 17-32.
11	Ibrahim JG, Chen MH, and Sinha D (2001). Bayesian Survival Analysis, Springer, New York.
12	Kirkwood JM, Ibrahim JG, Sondak VK, et al. (2000). High- and low-dose interferon alfa-2b in high-risk melanoma: ﬁrst analysis of intergroup trial E1690/S9111/C9190, Journal of Clinical Oncology, 18, 2444-2458. DOI
13	Li CS, Taylor JMG, and Sy JP (2001). Identiﬁability of cure models, Statistics & Probability Letters, 54, 389-395. DOI
14	Louzada F, Roman M, and Cancho VG (2011). The complementary exponential geometric distribu- tion: model, properties, and a comparison with its counterpart, Computational Statistics & Data Analysis, 55, 2516-2524. DOI
15	Maller RA and Zhou, X (1996). Survival Analysis with Long-Term Survivors, Wiley, New York.
16	Marshall AW and Olkin I (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, 84, 641-652. DOI
17	Ortega EMM, Cordeiro GM, and de Pascoa MAR (2011). The generalized Gamma Geometric distribution, Journal of Statistical Theory and Applications, 3, 433-454.
18	R Development Core Team (2013). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria.
19	Rigby RA and Stasinopoulos DM (2005). Generalized additive models for location, scale and shape (with discussion), Journal of the Royal Statistical Society. Series C (Applied Statistics), 54, 507-554. DOI
20	Smith RL and Naylor JC (1987). A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Journal of the Royal Statistical Society. Series C (Applied Statistics), 36, 358-369.
21	Lawless JF (2002). Statistical Models and Methods for Lifetime Data (2nd ed), Wiley, New York.
22	Stacy EW (1962). A generalization of the gamma distribution, The Annals of Mathematical Statistics, 33, 1187-1192. DOI
23	Tojeiro C, Louzada F, Roman M, and Borges P (2012). The complementary Weibull geometric distribution, Journal of Statistical Computation and Simulation, 84, 1345-1362.