The extended gamma distribution with regression model and applications

Emrah Altun; Mustafa Ç. Korkmaz; M. El-Morshedy; M. S. Eliwa; Emrah Altun; Mustafa Ç. Korkmaz; M. El-Morshedy; M. S. Eliwa

doi:10.3934/math.2021147

AIMS Mathematics

2021, Volume 6, Issue 3: 2418-2439. doi: 10.3934/math.2021147

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Research article

The extended gamma distribution with regression model and applications

1.
Department of Mathematics, Bartin University, Bartin, Turkey
2.
Department of Measurement and Evaluation, Artvin Çoruh University, Artvin, Turkey
3.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 24 September 2020 Accepted: 11 December 2020 Published: 18 December 2020
MSC : 62E15

This paper introduces a new extension of the gamma distribution, named as a new extended gamma distribution, via mixture representation of xgamma and gamma distributions. The statistical properties of the proposed distribution are derived such as moment generating and characteristic functions, variance, skewness, and kurtosis measures, Lorenz curve, and mean residual life function. The maximum likelihood, parametric bootstrap, method of moments, least squares, and weighted least squares estimation methods are considered to obtain the unknown model parameters. The finite sample performance of estimation methods is discussed via a simulation study. Using the proposed distribution, we propose a new regression model for the right-skewed response variable as an alternative to the gamma regression model. Two real data sets are analyzed to convince the readers for the usefulness of the proposed model.
- xgamma distribution,
- gamma distribution,
- maximum likelihood,
- method of moments,
- bias-correction,
- randomized quantile residuals
Citation: Emrah Altun, Mustafa Ç. Korkmaz, M. El-Morshedy, M. S. Eliwa. The extended gamma distribution with regression model and applications[J]. AIMS Mathematics, 2021, 6(3): 2418-2439. doi: 10.3934/math.2021147

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Abstract

This paper introduces a new extension of the gamma distribution, named as a new extended gamma distribution, via mixture representation of xgamma and gamma distributions. The statistical properties of the proposed distribution are derived such as moment generating and characteristic functions, variance, skewness, and kurtosis measures, Lorenz curve, and mean residual life function. The maximum likelihood, parametric bootstrap, method of moments, least squares, and weighted least squares estimation methods are considered to obtain the unknown model parameters. The finite sample performance of estimation methods is discussed via a simulation study. Using the proposed distribution, we propose a new regression model for the right-skewed response variable as an alternative to the gamma regression model. Two real data sets are analyzed to convince the readers for the usefulness of the proposed model.

References

[1]	B. S. Everitt, D. J. Hand, Finite Mixture Distributions, Chapman and Hall, London, 1981.
[2]	W. C. Chen, B. M. Hill, J. B. Greenhouse, J. V. Fayos, Bayesian analysis of survival curves for cancer patients following treatment, Bayesian Stat., 2 (1985), 299–328.
[3]	M. Erisoglu, N. Calis, T. Servi, U. Erisoglu, M. Topaksu, The mixture distribution models for interoccurence times of earthquakes, Russian Geol. Geophys., 52 (2011a), 685–692. doi: 10.1016/j.rgg.2011.06.001
[4]	U. Erisoglu, M. Erisoglu, H. Erol, A mixture model of two different distributions approach to the analysis of heterogeneous survival data, Int. J. Comput. Math. Sci. 5 (2011b), 75–79.
[5]	U. Erisoglu, M. Erisoglu, H. Erol, Mixture model approach to the analysis of heterogeneous survival data, Pakistan J. Stat., 5 (2012), 115–130.
[6]	K. E. Ahmad, A. M. Abd-El Rahman, Updating a nonlinear discriminant function estimated from a mixture of two Weibull distributions, Math. Comput. Model., 19 (1994), 41–51.
[7]	R. Jiang, D. N. P. Murthy, Two sectional models involving three Weibull distributions, Qual. Reliab. Eng. Int., 13 (1997), 83–96. doi: 10.1002/(SICI)1099-1638(199703)13:2<83::AID-QRE77>3.0.CO;2-V
[8]	K. S. Sultan, M. A. Ismail, A. S. Al-Moisheer, Mixture of two inverse Weibull distributions: Properties and estimation, Comput. Stat. Data Anal., 51 (2007), 5377–5387.
[9]	H. Zakerzadeh, A. Dolati, The generalized Lindley distribution, J. Math. Ext., 3 (2009), 1–17.
[10]	S. F. Ateya, Maximum likelihood estimation under a finite mixture of generalized exponential distributions based on censored data, Stat. Papers, 55 (2014), 311–325.
[11]	A. M. Abouammoh, A. M. Alshangiti, I. E. Ragab, A new generalized Lindley distribution, J. Stat. Comput. Simul., 85 (2015), 3662–3678. doi: 10.1080/00949655.2014.995101
[12]	A. H. El-Bassiouny, E. D. Medhat, M. Abdelfattah, M. S. Eliwa, Mixture of exponentiated generalized Weibull-Gompertz distribution and its applications in reliability, J. Stat. Appl. Probab., 5 (2016), 455–468. doi: 10.18576/jsap/050310
[13]	A. Karakoca, U. Erisoglu, M. Erisoglu, A comparison of the parameter estimation methods for bimodal mixture Weibull distribution with complete data, J. Appl. Stat., 42 (2015), 1472–1489. doi: 10.1080/02664763.2014.1000275
[14]	D. V. Lindley, Fiducial distributions and Bayes' theorem, J. Royal Stat. Society. Series B (Methodological), 20 (1958), 102–107.
[15]	M. E. Ghitany, D. K. Al-Mutairi, N. Balakrishnan, L. J. Al-Enezi, Power Lindley distribution and associated inference, Comput. Stat. Data Anal., 64 (2013), 20–33.
[16]	S. Nedjar, H. Zeghdoudi, On gamma Lindley distribution: Properties and simulations, J. Comput. Appl. Math., 298 (2016), 167–174. doi: 10.1016/j.cam.2015.11.047
[17]	S. Sen, S. S. Maiti, N. Chandra, The xgamma distribution: Statistical properties and application, J. Modern Appl. Stat. Methods, 15 (2016), 774–788. doi: 10.22237/jmasm/1462077420
[18]	M. Bourguignon, M. D. C. S. Lima, J. Leao, A. D. Nascimento, L. G. B. Pinho, G. M. Cordeiro, A new generalized gamma distribution with applications, Am. J. Math. Manage. Sci., 34 (2015), 309-342.
[19]	S. D. Waymyers, S. Dey, H. Chakraborty, A new generalization of the gamma distribution with application to negatively skewed survival data, Commun. Stat.-Simul. Comput., 47 (2018), 2083–2101. doi: 10.1080/03610918.2017.1335408
[20]	M. A. de Pascoa, E. M. Ortega, G. M. Cordeiro, The Kumaraswamy generalized gamma distribution with application in survival analysis, Stat. Methodol., 8 (2011), 411–433. doi: 10.1016/j.stamet.2011.04.001
[21]	M. O. Lorenz, Methods of measuring the concentration of wealth, Publ. Am. Stat. Assoc., 9 (1905), 209–219.
[22]	B. Efron, The jackknife, the bootstrap, and other resampling plans, Vol. 38. Philadelphia, PA, USA: SIAM, 1982.
[23]	J. Mazucheli, A. F. B. Menezes, S. Dey, Bias-corrected maximum likelihood estimators of the parameters of the inverse Weibull distribution, Comm. Stat.-Simul. Comput., 48 (2019), 2046–2055. doi: 10.1080/03610918.2018.1433838
[24]	J. Mazucheli, A. F. B. Menezes, S. Dey, Improved maximum-likelihood estimators for the parameters of the unit-gamma distribution, Comm. Stat.-Theory Methods, 47 (2018), 3767–3778. doi: 10.1080/03610926.2017.1361993
[25]	F. Ding, Decomposition based fast least squares algorithm for output error systems, Signal Process., 93 (2013), 1235–1242. doi: 10.1016/j.sigpro.2012.12.013
[26]	F. Ding, P. X. Liu, G. Liue, Gradient based and least-squares based iterative identification methods for OE and OEMA systems, Signal Process., 20 (2010), 664–677.
[27]	A. Zaka, A. S. Akhter, R. Jabeen, The new reflected power function distribution: Theory, simulation application., Aims Math., 5 (2020), 5031–5054.
[28]	S. M. Zaidi, M. M. A. Sobhi, M. El-Morshedy, A. Z. Afify, A new generalized family of distributions: Properties and applications, Aims Math., 6 (2021), 456–476. doi: 10.3934/math.2021028
[29]	E. Altun, H. M. Yousof, G. G. Hamedani, A new log-location regression model with influence diagnostics and residual analysis, Facta Universitatis, Series: Math. Inf., 33 (2018), 417–449.
[30]	E. C. Cuervo, Modelagem da variabilidade em modelos lineares generalizados, (Doctoral dissertation, Tese de D. Sc., IM-UFRJ, Rio de Janeiro, RJ, Brasil, 2001.
[31]	E. Cepeda, D. Gamerman, Bayesian methodology for modeling parameters in the two parameter exponential family, Revista Estadística, 57 (2005), 93–105.
[32]	T. F. Bateson, Gamma regression of interevent waiting times versus poisson regression of daily event counts: Inside the epidemiologist's toolbox-selecting the best modeling tools for the job, Epidemiology, 20 (2009), 202–204. doi: 10.1097/EDE.0b013e3181977688
[33]	E. Altun, Weighted-exponential regression model: An alternative to the gamma regression model, Int. J. Model., Simul., Sci. Comput., 10 (2019), 1–15.
[34]	E. Altun, The Lomax regression model with residual analysis, J. Appl. Stat., Forthcoming (2020), 1–10.
[35]	J. McCullagh, J. Nelder, Generalized Linear Models, Second Edition, Chapman and Hall, London, 1989.
[36]	P. K. Dunn, G. K. Smyth, Randomized quantile residuals, J. Comput. Graphical Stat., 5 (1996), 236–244.
[37]	P. Feigl, M. Zelen, Estimation of exponential survival probabilities with concomitant information, Biometrics, 21 (1965), 826–838. doi: 10.2307/2528247
[38]	M. E. Mead, The beta exponentiated Burr XⅡ distribution, J. Stat.: Advances Theory Appl. 12 (2014), 53–73.
[39]	M. V. Aarset, How to identify a bathtub hazard rate, IEEE Trans. Reliab., 36 (1987), 106–108.

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