Abstract
In this paper, we introduce a new continuous probability distribution with five parameters called the modified beta Gompertz distribution. It is derived from the modified beta generator proposed by Nadarajah, Teimouri and Shih (2014) and the Gompertz distribution. By investigating its mathematical and practical aspects, we prove that it is quite flexible and can be used effectively in modeling a wide variety of real phenomena. Among others, we provide useful expansions of crucial functions, quantile function, moments, incomplete moments, moment generating function, entropies and order statistics. We explore the estimation of the model parameters by the obtained maximum likelihood method. We also present a simulation study testing the validity of maximum likelihood estimators. Finally, we illustrate the flexibility of the distribution by the consideration of two real datasets.
Highlights
The Gompertz distribution is a continuous probability distribution introduced by Gompertz [1].The literature about the use of the Gompertz distribution in applied areas is enormous
A feature of the Gompertz distribution is that g( x ) is unimodal and has positive skewness, whereas the related hazard rate function given by Mathematics 2019, 7, 3; doi:10.3390/math7010003
It is based on the modified beta generator developed by Nadarajah et al [20]
Summary
The Gompertz distribution is a continuous probability distribution introduced by Gompertz [1]. From a mathematical point of view, the cumulative probability density function (cdf) of the Gompertz distribution with parameters λ > 0 and α > 0 is given by. The related probability density function (pdf) is given by g( x ) = λeαx e− α (e λ αx −1) It can be viewed as a generalization of the exponential distribution (obtained with α → 0) and an alternative to the gamma or Weibull distribution. The related applications show that a plays an important role in term of model flexibility This idea was extended by Jafari et al [4] who used the so-called beta generator introduced by Eugene et al [5]. We present and study a distribution with five parameters extending the Gompertz distribution It is based on the modified beta generator developed by Nadarajah et al [20].
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