A Distribution - Free Goodness of Fit Test for Copula Model: An Application to Farlie–Gumbel–Morgernstern Copula

With the advent of the concept of ‘Big Data’, the past couple of decades have witnessed a substantial rise in the application of probability models other than the multivariate normal model in dealing with multidimensional data. This phenomenon has been observed in various research fields ranging from finance to ecology, from gene expression to social media, etc. Copula has been the prima facie alternative tool to the widely and often inappropriately used multivariate normal model. But with the inception of copula theory we have been presented with a variety of copula-based non-normal models. This naturally leads to the inevitable question of the goodness of fit tests of these non-normal models for a given dataset. In this work, we have tried to address this question by developing a novel data driven goodness of fit (GoF) test for the Farlie–Gumbel–Morgenstern copula-based bivariate distribution. This approach can also act as a template for other copula-based distributions. Further, our proposed family of parametric bootstrap GoF tests does not require the knowledge of any sampling distribution, and works through a set of computational steps which can be implemented easily. We have studied the performance of the proposed GoF tests in terms of size and power. Finally, we have applied the test on a real dataset from Mekong River delta (in Vietnam) pertaining to arsenic contamination in groundwater.