On the Estimation of Empirical Copulas and Joint Densities

Abstract

Extended Abstract Copulas are principally utilized for modelling dependencies in multivariate distributions. The key idea behind copulas is that the joint distribution of two or more variables can be represented in terms of their marginal distributions and a specific correlation structure. As a measure of dependence, they have for instance found applications in reliability theory, signal processing, geodesy, hydrology and medicine. Results involving empirical bivariate copula densities are discussed in this presentation. In the proposed methodology, kernel density estimates are utilized to determine the marginal distributions. Then, a moment-based approximation technique for estimating a copula density from bivariate observations is introduced. This approach relies on the estimated joint density, the marginal densities and polynomial representations of the inverse marginal distribution functions. The joint density can be determined by multiplying a base density by a bivariate polynomial whose coefficients are obtained from the joint moments of the distributions. The degrees of the bivariate polynomial adjustment will be selected according to a certain goodness-of-fit criterion. The resulting simple representation of this copula density is suitable for reporting purposes or carrying out further algebraic manipulation. A new approach for obtaining an initial copula density estimate from Deheuvels’ empirical copula is also proposed. For a given bivariate data set, Deheuvels’ copula is evaluated and approximated by means of a bivariate least-squares polynomial. On differentiating this polynomial estimate with respect to both variables, one can obtain a preliminary estimate of the copula density function. Additionally, of a joint density function from the support of a density estimate of the standardized vector, which is taken to be a rectangle. turns out the domain of the original distribution can be delimited by a parallelogram once the back transformation is applied. joint density a