The Odd Log-Logistic Generalized Gompertz Distribution: Properties, Applications and Different Methods of Estimation

Abstract

We introduce a four-parameter lifetime distribution called the odd log-logistic generalized Gompertz model to generalize the exponential, generalized exponential and generalized Gompertz distributions, among others. We obtain explicit expressions for the moments, moment-generating function, asymptotic distribution, quantile function, mean deviations and distribution of order statistics. The method of maximum likelihood estimation of parameters is compared by six different methods of estimations with simulation study. The applicability of the new model is illustrated by means of a real data set.

Keywords

Odd log-logistic family of distribution / Maximum likelihood estimators / Least squares estimators / Weighted least squares estimators / Method of maximum product spacing / Percentile estimators

References

[1] Aarset AS. How to identify a bathtub hazard rate. IEEE Trans. Reliab.. 1987, 36 106-108

[2] Anderson TW, Darling DA. Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. Ann. Math. Stat.. 1952, 23 193-212

[3] Bemmaor AC, Glady N. Modeling purchasing behavior with sudden “death”: a flexible customer lifetime model. Manag. Sci.. 2012, 58 1012-1021

[4] Benkhelifa L. The beta generalized Gompertz distribution. Appl. Math. Model.. 2017, 52 341-357

[5] Benkhelifa L. The Marshall-Olkin extended generalized Gompertz distribution. J. Data Sci.. 2017, 15 227-254

[6] Cheng, R.C.H., Amin, N.A.K.: Maximum product-of-spacings estimation with applications to the lognormal distribution. Technical Report, Department of Mathematics, University of Wales (1979)

[7] Cheng RCH, Amin NAK. Estimating parameters in continuous univariate distributions with a shifted origin. J. R. Stat. Soc. Ser. B (Methodological). 1983, 45 394-403

[8] Choi K, Bulgren WG. An estimation procedure for mixtures of distributions. J. R. Stat. Soc. Ser. B (Methodological). 1968, 30 444-460

[9] da Silva RC, Sanchez JJD, Lima FP, Cordeiro GM. The Kumaraswamy Gompertz distribution. J. Data Sci.. 2015, 13 241-260

[10] Dey S, Mazucheli J, Nadarajah S. Kumaraswamy distribution: different methods of estimation. Comput. Appl. Math.. 2018, 37 2094-2111

[11] Economos AC. Rate of aging, rate of dying and the mechanism of mortality. Arch. Gerontol. Geriatr.. 1982, 1 46-51

[12] El-Gohary A, Al-Otaibi AN. The generalized Gompertz distribution. Appl. Math. Model.. 2013, 37 13-24

[13] Gradshteyn IS, Ryzhik IM. Table of Integrals, Series and Products. 2000 New York: Academic Press

[14] Gompertz B. On the nature of the function expressive of the law of human mortality and on the new mode of determining the value of life contingencies. Philos. Trans. R. Stat. Soc.. 1825, 115 513-580

[15] Gleaton JU, Lynch JD. Properties of generalized log-logistic families of lifetime distributions. J. Probab. Stat. Sci.. 2006, 4 51-64

[16] Jafari AA, Tahmasebi S, Alizadeh M. The beta-Gompertz distribution. Rev. Colomb. Estad.. 2014, 37 141-158

[17] Merovci F, Puka L. Transmuted Pareto distribution. ProbStat Forum. 2014, 7 1-11

[18] Milgram M. The generalized integro-exponential function. Math. Comput.. 1985, 44 443-458

[19] Roozegar R, Tahmsebi S, Jafari AA. The McDonald Gompertz distribution: properties and applications. Commun. Stat. Simul. Comput.. 2017, 46 3341-3355

[20] Ohishi K, Okamura H, Dohi T. Gompertz software reliability model: estimation algorithm and empirical validation. J. Syst. Softw.. 2009, 82 535-543

[21] Swain JJ, Venkatraman S, Wilson JR. Least-squares estimation of distribution functions in Johnson’s translation system. J. Stat. Comput. Simul.. 1988, 29 271-297

[22] Willemse W, Koppelaar H. Knowledge elicitation of Gompertz’ law of mortality. Scand. Actuar. J.. 2000, 2 168-179