Generalization of Gompertz Distribution and its Applications in Reliability and Time series

Generalization of Gompertz Distribution

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

In this paper, as a generalization of the Gompertz distribution, Marshall-
Olkin Gompertz distribution is considered. A three parameter AR(1) process is also considered. When X and Y are two independent random variables following Marshall Olkin Gompertz distribution, then average bias, average mean square error, average confidence length and coverage probability of the of the simulated estimates of reliability R is computed. Data analysis based on a real data set and modeling are also done.