Absolutely Continuous Semi-parametric Bivariate Distributions

In this paper we have introduced a very flexible semi-parametric absolute continuous bivariate distribution based on proportional hazard class of distributions. Block and Basu’s bivariate exponential distribution is a member of this class. The key feature of the proposed class is that we do not assume any specific form of the base line distribution function, hence it becomes a very flexible class of distribution functions. We derive several properties of this class. We provide the classical inference of the unknown parameters. In this case it has been observed that the maximum likelihood estimators (MLEs) cannot be obtained in closed form. Moreover, it is observed that the standard gradient descent algorithm for this multi-dimensional optimization problem often does not converge to the optimal solution. We have proposed a very effective expectation maximization (EM) algorithm to compute the MLEs. It involves solving a lower dimensional optimization problem at each step, and the implementation of the proposed EM algorithm is quite simple. Extensive simulation results suggest that the EM algorithm converges and it converges close to the true values of the parameters. We have addressed goodness of fit test also in this case. One bivariate data set has been analyzed to show how the proposed model can be used in practice. The results are quite satisfactory.