Approximation Multivariate Distribution with pair copula Using the Orthonormal Polynomial and Legendre Multiwavelets basis functions

摘要

In this paper, we concentrate on new methodologies for copulas introduced anddeveloped by Joe, Cooke, Bedford, Kurowica, Daneshkhah and others on the newclass of graphical models called vines as a way of constructing higherdimensional distributions. We develop the approximation method presented byBedford et al (2012) at which they show that any $n$-dimensional copula densitycan be approximated arbitrarily well pointwise using a finite parameter set of2-dimensional copulas in a vine or pair-copula construction. Our constructiveapproach involves the use of minimum information copulas that can be specifiedto any required degree of precision based on the available data or experts'judgements. By using this method, we are able to use a fixed finite dimensionalfamily of copulas to be employed in a vine construction, with the promise of auniform level of approximation. The basic idea behind this method is to use a two-dimensional ordinarypolynomial series to approximate any log-density of a bivariate copula functionby truncating the series at an appropriate point. We present an alternativeapproximation of the multivariate distribution of interest by consideringorthonormal polynomial and Legendre multiwavelets as the basis functions. Weshow the derived approximations are more precise and computationally fasterwith better properties than the one proposed by Bedford et al. (2012). We thenapply our method to modelling a dataset of Norwegian financial data that waspreviously analysed in the series of papers, and finally compare our results bythem.