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About this book

This textbook provides a step-by-step introduction to the class of vine copulas, their statistical inference and applications. It focuses on statistical estimation and selection methods for vine copulas in data applications. These flexible copula models can successfully accommodate any form of tail dependence and are vital to many applications in finance, insurance, hydrology, marketing, engineering, chemistry, aviation, climatology and health.

The book explains the pair-copula construction principles underlying these statistical models and discusses how to perform model selection and inference. It also derives simulation algorithms and presents real-world examples to illustrate the methodological concepts. The book includes numerous exercises that facilitate and deepen readers’ understanding, and demonstrates how the R package VineCopula can be used to explore and build statistical dependence models from scratch. In closing, the book provides insights into recent developments and open research questions in vine copula based modeling.

The book is intended for students as well as statisticians, data analysts and any other quantitatively oriented researchers who are new to the field of vine copulas. Accordingly, it provides the necessary background in multivariate statistics and copula theory for exploratory data tools, so that readers only need a basic grasp of statistics and probability.

Table of Contents

Frontmatter

Chapter 1. Multivariate Distributions and Copulas

Abstract
Before we describe multivariate distribution, we review some notation and characteristics of univariate distributions. In general, we use capital letters for random variables and small letters for observed values, i.e., we write \(X=x\). Here, we only consider absolutely continuous or discrete distributions, and therefore corresponding (conditional) densities or probability mass functions exist. In both cases, we use the letter f for densities or probability mass functions and the letter F for the corresponding distribution function.
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Chapter 2. Dependence Measures

Abstract
There exist several measures for the strength and direction of dependence between two random variables. The most common ones are the Pearson product–moment correlation, Kendall’s tau, and Spearman’s rho. We give a short introduction to these measures including their estimation based on data. Kendall’s tau and Spearman’s rho can be expressed in terms of the corresponding copula alone, while this is not always the case for the Pearson product–moment correlation. For joint extreme events, the notion of tail dependence coefficients will be introduced. Finally, the concepts of partial and conditional correlations will be discussed, which will play an important role in the class of multivariate vine copulas.
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Chapter 3. Bivariate Copula Classes, Their Visualization, and Estimation

Abstract
There are three major construction approaches of copulas. One arising from applying the probability integral transform (see Definition 1.​3) to each margin of known multivariate distributions and one to use generator functions. The first approach applied to elliptical distributions yields the class of elliptical copulas. With the second approach, we obtain the class of Archimedean copulas. The well-known examples of this class are the Clayton, Gumbel, Frank, and Joe copula families. The third approach arises from extensions of univariate extreme-value theory to higher dimensions.
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Chapter 4. Pair Copula Decompositions and Constructions

Abstract
The goal is to construct multivariate distributions using only bivariate building blocks. The appropriate tool to obtain such a construction is to use conditioning. (Rüschendorf, Schweizer and Taylor (Ed.), Distributions with Fixed Marginals and Related Topics, 1996) Joe (1996) gave the first pair copula construction of a multivariate copula in terms of distribution functions, while (Proceedings of ESREL2001. Turin, Italy, 2001) Bedford and Cooke (2001), (Annals of Statistics, 30(4):1031–1068, 2002) Bedford and Cooke (2002) independently developed constructions expressed in terms of densities. Additionally they provided a general framework to identify all possible constructions.
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Chapter 5. Regular Vines

Abstract
In the last chapter we saw that we can construct trivariate distributions using only bivariate building blocks. Additionally we introduced special vine distribution classes such as C- and D-vine distributions in arbitrary dimensions through recursive conditioning. In this chapter we generalize this construction principle allowing for different conditioning orders. As we already noted the construction is not unique, therefore it will be important to allow for different constructions and to organize them.
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Chapter 6. Simulating Regular Vine Copulas and Distributions

Abstract
For simulation from a d-dimensional distribution function \(F_{1,..., d}\) with conditional distribution functions \(F_{j|1,\ldots , j-1}(\cdot |x_1,\ldots , x_{j-1})\) and their inverses \(F_{j|1,\dots , j-1}^{-1}(\cdot |x_1,\ldots , x_{j-1})\) for \(j=2,\ldots , d\) we can use iterative inverse probability transformations.
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Chapter 7. Parameter Estimation in Simplified Regular Vine Copulas

Abstract
In the last chapter, we have seen how to design very flexible multivariate copulas. To make them useful for practise, we have now to tackle the problem of parameter estimation. For this, we will assume that the simplifying assumption holds throughout. We also suppose, that the vine copula model is completely specified by a given vine tree sequence, as well as, the pair copula families associated to each edge in the vine tree sequence. The problem of how to choose the pair copula families and how to select the vine structure will be treated in subsequent chapters.
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Chapter 8. Selection of Regular Vine Copula Models

Abstract
The full specification of a vine copula model requires the choice of a vine tree structure, copula families for each pair copula term and their corresponding parameters. In this chapter, we discuss different frequentist selection and estimation approaches. The three-layered definition of a regular vine copula leads to three fundamental estimation and selection tasks:
1.
Estimation of copula parameters for a chosen vine tree structure and chosen pair copula families for each edge in the vine tree structure,
 
2.
Selection of the parametric copula family for each pair copula term and estimation of the corresponding parameters for a chosen vine tree structure,
 
3.
Selection and estimation of all three model components.
 
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Chapter 9. Comparing Regular Vine Copula Models

Abstract
In this chapter, we want to compare the fit of two or more regular vine copula specifications for a given copula data set.
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Chapter 10. Case Study: Dependence Among German DAX Stocks

Abstract
Understanding dependence among financial stocks is vital for option pricing and forecasting portfolio returns. Copula modeling has a long history in this area. However, the restrictive use of the multivariate Gaussian copula with no tail dependence has been blamed for the financial crisis in 2007–2008 (see Li 2000, Salmon 2012 and recently Puccetti and Scherer 2018), and therefore it is important to allow for much more flexible dependence models such as allowed by the vine copula class. In this context, the possibility of modeling tail dependence by vine copulas has to be mentioned.
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Chapter 11. Recent Developments in Vine Copula Based Modeling

Abstract
So far, we have given a basic introduction to vine copulas with some applications. The field is rapidly developing and in this chapter we showcase some developments in the area of estimation, model selection, and adaptions to special data structures. Next, we summarize current applications to problems from finance, life and earth sciences. The chapter closes with a section on available software to conduct vine copula based modeling.
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Backmatter

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