Multivariate patchwork copulas: a unified approach with applications to partial comonotonicity

none2siWe present a general view of patchwork constructions of copulas that encompasses previous approaches based on similar ideas (ordinal sums, gluing methods, piecing-together, etc.). Practical applications of the new methodology are connected with the determination of copulas having specified behaviour in the tails, such as upper comonotonic copulas.Durante, Fabrizio; Sánchez, Juan Fernández; Sempi, CarloDurante, Fabrizio; Sánchez, Juan Fernández; Sempi, Carl

Copulas with given values on a horizontal and a vertical section

summary:In this paper we study the set of copulas for which both a horizontal section and a vertical section have been given. We give a general construction for copulas of this type and we provide the lower and upper copulas with these sections. Symmetric copulas with given horizontal section are also discussed, as well as copulas defined on a grid of the unit square. Several examples are presented

A new family of trivariate proper quasi-copulas

summary:In this paper, we provide a new family of trivariate proper quasi-copulas. As an application, we show that $W^{3}$ – the best-possible lower bound for the set of trivariate quasi-copulas (and copulas) – is the limit member of this family, showing how the mass of $W^3$ is distributed on the plane $x+y+z=2$ of $[0,1]^3$ in an easy manner, and providing the generalization of this result to $n$ dimensions

Generalized Additive Modeling For Multivariate Distributions

In this thesis, we develop tools to study the influence of predictors on multivariate distributions. We tackle the issue of conditional dependence modeling using generalized additive models, a natural extension of linear and generalized linear models allowing for smooth functions of the covariates. Compared to existing methods, the framework that we develop has two main advantages. First, it is completely flexible, in the sense that the dependence structure can vary with an arbitrary set of covariates in a parametric, nonparametric or semiparametric way. Second, it is both quick and numerically stable, which means that it is suitable for exploratory data analysis and stepwise model building. Starting from the bivariate case, we extend our framework to pair-copula constructions, and open new possibilities for further applied and methodological work. Our regression-like theory of the dependence, being built on conditional copulas and generalized additive models, is at the same time theoretically sound and practically useful

Approximating non-Gaussian Bayesian networks using minimum information vine model with applications in financial modelling

Many financial modeling applications require to jointly model multiple uncertain quantities to presentmore accurate, near future probabilistic predictions. Informed decision making would certainly benefitfrom such predictions. Bayesian networks (BNs) and copulas are widely used for modeling numerousuncertain scenarios. Copulas, in particular, have attracted more interest due to their nice property ofapproximating the probability distribution of the data with heavy tail. Heavy tail data is frequentlyobserved in financial applications. The standard multivariate copula suffer from serious limitations whichmade them unsuitable for modeling the financial data. An alternative copula model called the pair-copulaconstruction (PCC) model is more flexible and efficient for modeling the complex dependence of finan-cial data. The only restriction of PCC model is the challenge of selecting the best model structure. Thisissue can be tackled by capturing conditional independence using the Bayesian network PCC (BN-PCC).The flexible structure of this model can be derived from conditional independences statements learnedfrom data. Additionally, the difficulty of computing conditional distributions in graphical models for non-Gaussian distributions can be eased using pair-copulas. In this paper, we extend this approach furtherusing the minimum information vine model which results in a more flexible and efficient approach inunderstanding the complex dependence between multiple variables with heavy tail dependence andasymmetric features which appear widely in the financial applications

Nonparametric Bayes modeling of count processes

Data on count processes arise in a variety of applications, including longitudinal, spatial and imaging studies measuring count responses. The literature on statistical models for dependent count data is dominated by models built from hierarchical Poisson components. The Poisson assumption is not warranted in many applications, and hierarchical Poisson models make restrictive assumptions about over-dispersion in marginal distributions. This article proposes a class of nonparametric Bayes count process models, which are constructed through rounding real-valued underlying processes. The proposed class of models accommodates applications in which one observes separate count-valued functional data for each subject under study. Theoretical results on large support and posterior consistency are established, and computational algorithms are developed using Markov chain Monte Carlo. The methods are evaluated via simulation studies and illustrated through application to longitudinal tumor counts and asthma inhaler usage

Tensor approximation of generalized correlated diffusions and applications

This thesis documents my research activity conducted in the past three years at the Department of Statistical Science at the University College London. My investigation is focused on functional-analytic methods applied to the characterization of generalized correlated Markov processes. The main objective of the research is to formalize the properties of such a class of stochastic processes when approximated in a tensor space. This lead to the development of a new interpretation of the correlation among processes that is exploited for the analysis of copula functions and their statistical properties