Advances in Copula Estimation and Distribution Theory
Abstract
This dissertation initially features distributional results related to copulas. Four distinct copula density estimation methodologies, including Bernstein’s polynomial approximation, are proposed and criteria for the selection of their tuning parameters are provided. These four approaches were found to produce similar density estimates, which validates their suitability. Moreover, the copula associated with a Wiener process and its running maximum is determined, and an illustrative numerical example is presented. The principal properties of Spearman’s, Kendall’s, Blomqvist’ and Hoeffding’s measures of association as well as their representations in terms of copulas are also discussed. Then, a novel method that is based on an arctangent transformation is introduced for classifying the tail behaviour of continuous probability laws. The resulting categories prove consistent with those obtained by applying available criteria. As well, approximations to the distributions of quadratic forms in gamma, inverse Gaussian, binomial and Poisson random variables, which hinge on the determination of their moments via a symbolic approach, are proposed and several applications are pointed out. Additionally, an accurate density approximation that relies of an extension of the generalized gamma distribution is introduced and the case of quadratic forms in Hermitian matrices in complex Gaussian vectors is also addressed. Finally, a methodology involving the use of Fritsch-Carlson monotonic interpolating splines and the Kulback-Leibler measure of divergence is proposed for quantifying the proportion of information that is contained in collections of distributional moments.