The Odd Log-Logistic Generalized Gompertz Distribution: Properties, Applications and Different Methods of Estimation
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The Odd Log-Logistic Generalized Gompertz Distribution: Properties, Applications and Different Methods of Estimation Morad Alizadeh1
· Lazhar Benkhelifa2 · Mahdi Rasekhi3 · Bistoon Hosseini4
Received: 4 June 2018 / Revised: 4 August 2018 / Accepted: 27 December 2018 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2019
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 Mathematics Subject Classification 60E05 · 62E15 · 62F10
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Lazhar Benkhelifa [email protected] Morad Alizadeh [email protected] Mahdi Rasekhi [email protected] Bistoon Hosseini [email protected]
1
Department of Statistics, Persian Gulf University, Bushehr, Iran
2
Department of Mathematics and Informatics, Larbi Ben M’Hidi University, Oum El Bouaghi, Algeria
3
Department of Statistics, Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran
4
Kermanshah Province Electricity Distribution Company, Kermanshah, Iran
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M. Alizadeh et al.
1 Introduction A random variable X is said to have the generalized Gompertz distribution (GG), for x > 0, with parameters a, b, c > 0 if its cumulative distribution function(cdf) is given by G (x) = 1 − e
−b a x a (e −1)
c
,
(1.1)
where a, b > 0 are the scale parameters and c > 0 is the shape parameter. The probability density function (pdf) corresponding to (1.1) is g (x) = cbea x e
−b a x a (e −1)
c−1 −b a x 1 − e a (e −1) .
(1.2)
The GG distribution with parameters a, b and c will be denoted by GG (a, b, c). When c = 1, the GG distribution reduces to the Gompertz distribution. The Gompertz distribution, proposed by Gompertz [14], has been used to describe human mortality and establish actuarial tables. It has been also used in several research areas such as biology [11], sociology [22], computer [20] and marketing [3]. However, its hazard function can only be increasing or constant and thus cannot provide reasonable fits for modeling phenomenon with a bathtub-shaped hazard rate function which is an important feature for reliability studies. Due to this reason, generalizations of the Gompertz distribution have been proposed in the literature. El-Gohary et al. [12] proposed the generalized Gompertz dist
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