On the Marshall-Olkin extended distributions

A general method of introducing a new parameter to a well-established continuous baseline cumulative function G to obtain more flexible distributions was proposed by Marshall and Olkin (1997). This new family is known as Marshall-Olkin extended G family of distributions. In this article, we characterize this family as mixtures of the distributions of the minimum and maximum of random variables with cumulative function G. We demonstrate that the coefficients of the mixtures are probabilities of random variables with geometric distributions. Additionally, we present new representations for the density and cumulative functions of this class of distributions. Further, we introduce a new three-parameter continuous model for modeling rates and proportions based on the Marshall-Olkin's method. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of a real dataset.