Inferences for the Bivariate Probability Distribution Using Farlie - Gumbel -Morgenstern Copula

Abstract

This research addresses four major objectives in modelling a bivariate dataset, along with subsequent inferences, when a bivariate normal model is not tenable. The Farlie-Gumbel-Morgenstern Copula (FGMC) combines two marginals with a specific known functional form to obtain a class of distributions (henceforth known FGMD) indexed by an association parameter.

The first objective of our research has been the inferences on the association parameter based on a random sample from the FGMD. Several point estimators have been proposed. Sampling distributions as well as the bias and the mean squared error (MSE) of these estimators have been studied through extensive simulation which demonstrates the superiority of the Bayes estimators over a large segment of the parameter space.

The second objective is to check the independence of the variables in FGMD. It has been shown through simulation that the two most common asymptotic tests are simply not reliable enough (in terms of maintaining the size) for the underlying FGMD for small to moderately large sample sizes whereas, our proposed group of four tests based on the likelihood ratio (LR) principle, works remarkably well in terms of size and power even for very small sample sizes.

The third objective is the prediction. In this regard, it has been demonstrated how the proposed FGMD can be applicable in modelling groundwater quality data available from the Mekong Delta Region (MDR) where the presence of Arsenic is a major public health hazard. The values of other covariates can be used to estimate the level of As through the three FGMD based predictors.

The fourth objective has been to come up with a reliable GoF test for the FGMD. Our proposed four GoF tests use a simple and implementable PB approach, and finds its critical value through a series of computational steps. Through simulation we have demonstrated the utility of one of these tests which is computationally the least time consuming.

This work of ours can be used as a template to study copula based joint distribution which we have suggested at the end along with some directions to multivariate generalization of this bivariate FGMD.