Absolutely Continuous Bivariate Modified Weibull Distribution: Properties, Estimation

The Block and Basu bivariate exponential (BBBE) distribution is one of the most popular and widely used absolutely continuous bivariate distributions. Kundu and Gupta \cite{kundu2010class} obtained the Block-Basu bivariate Weibull (BBBW) distribution. Extensive work has been done on the BBBW model over the past several decades. Interestingly, it is observed that the BBBW model can be extended to the modified Weibull model also. We call this new model as the Block and Basu bivariate modified Weibull (BBBMW) distribution. We consider the properties of the BBBMW distribution and provide the associated copula function. The BBBMW model has five unknown parameters and the maximum likelihood estimators (MLEs) cannot be obtained in closed form. To compute the MLEs directly, one needs to solve a five-dimensional optimization problem. We propose to use the EM algorithm for computing the MLEs of the unknown parameters. The proposed EM algorithm can be carried out by solving a two-dimensional optimization problem at each EM step. An extensive simulation is carried out, which demonstrates that the proposed EM algorithm performs quite well. A real data set is analyzed for illustrative purposes.