Simplified Pair Copula Constructions --- Limits and Extensions

So called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep them tractable for inference and model selection, the simplifying assumption that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on is popular. In this paper, we show for which classes of distributions such a simplification is applicable, significantly extending the discussion of Hob{\ae}k Haff et al. (2010). In particular, we show that the only Archimedean copula in dimension d \geq 4 which is of the simplified type is that based on the gamma Laplace transform or its extension, while the Student-t copula is the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on

A Class of Models for Uncorrelated Random Variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence

Extremal t processes: Elliptical domain of attraction and a spectral representation

The extremal t process was proposed in the literature for modeling spatial extremes within a copula framework based on the extreme value limit of elliptical t distributions (Davison, Padoan and Ribatet (2012)). A major drawback of this max-stable model was the lack of a spectral representation such that for instance direct simulation was infeasible. The main contribution of this note is to propose such a spectral construction for the extremal t process. Interestingly, the extremal Gaussian process introduced by Schlather (2002) appears as a special case. We further highlight the role of the extremal t process as the maximum attractor for processes with finite-dimensional elliptical distributions. All results naturally also hold within the multivariate domain

Copula-Based Dependence Characterizations and Modeling for Time Series

This paper develops a new unified approach to copula-based modeling and characterizations for time series and stochastic processes. We obtain complete characterizations of many time series dependence structures in terms of copulas corresponding to their finite-dimensional distributions. In particular, we focus on copula- based representations for Markov chains of arbitrary order, m-dependent and r-independent time series as well as martingales and conditionally symmetric processes. Our results provide new methods for modeling time series that have prescribed dependence structures such as, for instance, higher order Markov processes as well as non-Markovian processes that nevertheless satisfy Chapman-Kolmogorov stochastic equations. We also focus on the construction and analysis of new classes of copulas that have flexibility to combine many different dependence properties for time series. Among other results, we present a study of new classes of cop- ulas based on expansions by linear functions (Eyraud-Farlie-Gumbel-Mongenstern copulas), power functions (power copulas) and Fourier polynomials (Fourier copulas) and introduce methods for modeling time series using these classes of dependence functions. We also focus on the study of weak convergence of empirical copula processes in the time series context and obtain new results on asymptotic gaussianity of such processes for a wide class of beta mixing sequences.

Skewed Factor Models Using Selection Mechanisms

Traditional factor models explicitly or implicitly assume that the factors follow a multivariate normal distribution; that is, only moments up to order two are involved. However, it may happen in real data problems that the first two moments cannot explain the factors. Based on this motivation, here we devise three new skewed factor models, the skew-normal, the skew-t, and the generalized skew-normal factor models depending on a selection mechanism on the factors. The ECME algorithms are adopted to estimate related parameters for statistical inference. Monte Carlo simulations validate our new models and we demonstrate the need for skewed factor models using the classic open/closed book exam scores dataset