Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions

Abstract

In this paper, we concentrate on new methodologies for copulas introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah and others on the new class of graphical models called vines as a way of constructing higher dimensional distributions. We develop the approximation method presented by Bedford et al (2012) at which they show that any n-dimensional copula density can be approximated arbitrarily well pointwise using a finite parameter set of 2-dimensional copulas in a vine or pair-copula construction. Our constructive approach involves the use of minimum information copulas that can be specified to 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 dimensional family of copulas to be employed in a vine construction, with the promise of a uniform level of approximation.The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We present an alternative approximation of the multivariate distribution of interest by considering orthonormal polynomial and Legendre multiwavelets as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed by Bedford et al. (2012). We then apply our method to modelling a dataset of Norwegian financial data that was previously analysed in the series of papers, and finally compare our results by them.

Probabilistic Modeling of Financial Uncertainties

“…starting with simple bases, and moving to more complex ones, and including them until we are satisfied with our approximation. Our OP basis functions are as follows, (Daneshkhah et al, 2012) Following the explanations to select basis function in an optimal manner, we add the basis functions by using stepwise method outlined in Daneshkhah et al (2012). In this method, at each phase, log-likelihood of adding each additional basis function is assessed.…”

“…Subsequently, function that develops the largest increase in the log-likelihood is included. Additionally, to obtain optimal results, first four bases should be considered (Daneshkhah et al, 2012).…”

“…The approach for unconditional copulas 46 , 34 , 45 is similar. Basis functions could be chosen based on the method described in Daneshkhah et al (2012) i.e. starting with simple bases, and moving to more complex ones, and including them until we are satisfied with our approximation.…”

“…It is straightforward to implement, with the only technical assumption of considering the copula density as continuous and non-zero. An appropriate approach to build an entropy copula or specifying dependency constraints is by moments (Daneshkhah et al, 2012). Moments can be specified either on the copula or on the underlying bivariate density.…”

Approximation Multivariate Distribution of Main Indices of Tehran Stock Exchange with Pair-Copula

“…The approach used herein allows a lot of flexibility in copula specification that was analyzed and some properties of it was said in Bedford et al (2013) and developed by Daneshkhah et al (2013), and for approximation multivariate distribution, a graphical model, called a vine, is used to systematically specify how two-dimensional copulas are stacked together to produce an -dimensional copula.…”