A Bivariate Exponential Distribution with q-Exponential Marginals and its Applications

Abstract

Bivariate Gumbel’s exponential distribution is one of the most popular continuous bivariate distributions. Comprehensive studies have been done on bivariate Gumbel’s exponential model during the past few decades. In this paper, we have derived a generalized version of bivariate Gumbel’s exponential model through entropy optimization and we call this model as q-bivariate Gumbel’s exponential model. One of the major properties of the q-bivariate Gumbel’s exponential model is that its marginal densities are q-exponential distributions. Its survival function, distribution function and density function can be expressed in terms of q-exponential function, which is the q-analogue of exponential function which posses several applications in various fields. Different properties and a characterisation theorem of this distribution have been discussed. For illustrating the use of the proposed model the unknown parameters are estimated using the method of maximum likelihood estimation. A likelihood ratio test is carried out to test the goodness of fit of q-bivariate Gumbel’s exponential distribution to verify its compatibility with the existing bivariate Gumbel’s exponential model. In order to interpret the practical applicability of q-bivariate Gumbel’s exponential model a simulation study and a real data application have been carried out. From this study, we can conclude that q-bivariate Gumbel’s exponential model shows a better fit than bivariate Gumbel’s exponential model.