Abstract
A new five parameter gamma-generalized modified Weibull (GGMW) distribution which includes exponential, Rayleigh, modified Weibull, Weibull, gamma-modified Weibull, gamma-modified Rayleigh, gamma-modified exponential, gamma-Weibull, gamma-Rayleigh, and gamma-exponential distributions as special cases is proposed and studied. Some mathematical properties of the new class of distributions including moments, distribution of the order statistics, and Renyi entropy are presented. Maximum likelihood estimation technique is used to estimate the model parameters and applications to a real datasets to illustrates the usefulness of the proposed class of models are presented.
Highlights
Weibull distribution has been widely used for modeling data in a wide variety of areas including reliability, engineering, stochastic processes, survival analysis and renewal theory
This class of distributions is flexible in accommodating all forms of hazard rate functions and contains several well known and new sub-models such as Weibull, Rayleigh, exponential, modified Weibull, gamma-modified Weibull, gamma-modified exponential, gamma-Weibull, gamma-Rayleigh, gamma-linear failure rate, gamma-extreme value, gamma-additive exponential and gamma-exponential distributions
There are several extensions of the Weibull distribution and its sub-models including the exponentiated Weibull (Mudholkar, Srivastava, and Kollia 1996), which is a special case of the beta Weibull distribution proposed by (Lee, Famoye, and Olumolade 2007), generalized Rayleigh (Kundu and Rakab 2005), exponentiated exponential (Gupta and Kundu 1999), (Gupta and Kundu 2001), modified Weibull (Mudholkar, Srivastava, and Friemer 1995), exponentiated modified Weibull (Sarhan and Zaindin 2009), and a host of other distributions, some of which are presented in section 2 of this paper
Summary
Weibull distribution has been widely used for modeling data in a wide variety of areas including reliability, engineering, stochastic processes, survival analysis and renewal theory. We present and study the mathematical properties of the gamma-generalized modified Weibull distribution. The four parameter generalizations include the additive Weibull distribution of (Xie and Lai 1995), modified Weibull (Sarhan and Zaindin 2009), beta-Weibull proposed by (Famoye et al 2005) and Kumaraswamy Weibull by (Cordeiro, Ortega, and Nadarajah 2010). We will consider and present a generalization of the generalized modified Weibull distribution via the family of distributions given in equation (5). In addition to the motivations provided by (Zografos and Balakrishnan 2009), we are interested in the generalization of the generalized modified Weibull distribution via the gamma-generator and establishing the relationship between weighted distributions and equations (1) and (5), respectively.
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