使用R进行Copula建模基础

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"《Elements of Copula Modeling with R》是一本详细介绍如何在R环境中使用copula包进行多元连续分布统计建模的专业书籍。作者包括Marius Hofert、Ivan Kojadinovic、Martin Mächler和Jun Yan，涵盖了风险管理、保险、金融、工程等多个领域的应用。" Copula理论是统计学中用于描述随机变量之间依赖关系的一种方法，它通过将标准均匀边际分布的多元分布函数分离出来，使得我们能够独立地处理各个变量的边缘分布，同时保留它们之间的依赖结构。这本书深入浅出地介绍了copula的主要理论发现，并提供了使用R语言及copula包进行实际建模的步骤。 Copula建模的核心在于它能够灵活地捕捉不同类型的依赖性，包括对称和非对称的、线性和非线性的，以及强依赖和弱依赖关系。在风险管理中，这种能力对于理解资产之间的关联性至关重要，特别是在计算风险度量如Value at Risk (VaR)时。在保险业，copula模型被用于估计理赔之间的关联，以更准确地评估潜在损失的风险。在金融领域，它们被用于构建复杂的金融衍生品模型。本书的作者团队由来自不同学术背景的专家组成，他们在统计学、保险科学、应用数学等领域有着深厚的研究基础。书中不仅涵盖了copula的基础理论，还涉及了多种类型的copula函数，如Frank、Gumbel、 Clayton、Joe等，以及如何通过似然比检验、AIC或BIC等准则选择合适的copula类型。此外，书中还讲解了如何估计copula参数，以及如何在R中实现这些操作。《Elements of Copula Modeling with R》还讨论了实证分析的案例，涵盖了各种实际数据集的应用，这有助于读者将理论知识应用于实践。这些案例可能来自于金融市场的收益率序列、天气数据、水文学记录等，展示了copula模型在不同领域的广泛适用性。这本书对于希望深入理解和应用copula理论的统计学家、数据科学家、金融分析师以及相关领域的专业人士来说，是一份宝贵的参考资料。通过学习本书，读者将能够掌握使用R语言进行copula建模的技能，从而更好地理解和管理多元数据中的复杂依赖结构。

6 1 Introduction

extreme-value theory: In that case, it is often more natural to standardize to unit

Fréchet margins with df F(x) = exp(−1/x), x>0. The important message is

that no matter what continuous univariate distribution the margins are transformed

to, it does not alter the philosophy behind the “copula approach” to the study of

dependence. Due to the probability and quantile transformations (see Lemmas 1.2.1

and 1.2.2, respectively), the choice of U(0, 1) margins turns out to be a natural and

convenient one.

In recent years, copulas have turned out to be the subject of a large number

of scientiﬁc publications; see, for instance, Genest et al. (2009) for a bibliometric

overview in ﬁnance. They were applied in a wide variety of areas such as quantita-

tive risk management, econometric modeling, or environmental modeling, to name

a very few; see, for example McNeil et al. (2015), Patton (2013), and Salvadori et al.

(2007). The reason for what could be inelegantly called “the copula craze” lies in

Sklar’s Theorem (which will be precisely stated in Sect. 2.3). Let (X

,...,X

) be a

d-dimensional random vector and let H(x) = P(X

≤ x

,...,X

≤ x

), x ∈ R

be its df. The ﬁrst part of Sklar’s Theorem asserts that every d-dimensional df H

can be expressed as

H(x) = C



),...,F

)



, x ∈ R

, (1.2)

in terms of a d-dimensional copula C and the univariate marginal dfs F

,...,F

obtained from H by F

) = H(∞,...,∞,x

, ∞,...,∞), x

∈ R. The copula

C is thus the function which connects or couples the marginal dfs F

,...,F

the multivariate df H , hence the name “copula.” For estimation of H from data, this

offers a great deal of ﬂexibility as it allows one to model the marginal dfs F

,...,F

separately from the dependence represented by the copula C, which is often of

interest from a statistical and numerical point of view. The second part of Sklar’s

Theorem provides a converse. Given any copula C and univariate dfs F

,...,F

a multivariate df H can be composed via (1.2) which then has univariate margins

,...,F

and “dependence structure” C. This is of interest in applications such as

stress testing in ﬁnance, insurance and quantitative risk management, and in model

building in general.

1.4 Structure and Philosophy of the Book

The book is organized as follows. Chapter 2 offers a basic introduction to copulas

and presents their main properties along with the most important theoretical

results such as the Fréchet–Hoeffding bounds, Sklar’s Theorem, and the invariance

principle. Chapter 3 introduces the main copula classes and the corresponding

sampling procedures, along with some copula transformations that are important

for practical purposes. The estimation of copulas is addressed in Chap. 4 from a

parametric, semi-parametric, and nonparametric perspective. Chapter 5 presents

graphical diagnostics, statistical tests and discusses model selection. Chapter 6 is

1.5 Additional Reading 7

concerned with more advanced topics such as the handling of ties, time series, and

covariates (in a regression-like setting). The appendix contains information about

the R session and package versions used throughout the book.

Overall, the theoretical concepts introduced in this book are illustrated by

numerous R examples using functions predominantly from the R package copula.

The examples and ﬁgures are stand-alone and reproducible. Most involve either

synthetic or real data.

The book was written according to our best knowledge. Errors are, however,

unavoidable in such a large project combining theoretical concepts and R code.

In particular, it may happen that some of the code in the book does not work as

expected anymore due to changes in one of the used R packages. An errata list is

available at http://copula.r-forge.r-project.org/book/ along with other information

on the book.

1.5 Additional Reading

As the title of the book indicates, only a small proportion of the existing research

on copulas is touched upon in this work. Besides the literature cited in the book and

on the help pages of the R package copula, the interested reader may consult one

of the following references on copula theory and applications; note that this list is

non-exhaustive and that some of the references are also cited later in this book at

appropriate places.

Early monographs on copulas are Joe (1997) (with focus on novel probabilistic

notions around copulas) and Nelsen (2006) (a well-known, readable introduction).

An interesting historical perspective and introduction can be found in Durante and

Sempi (2010). A more advanced probabilistic treatment of copulas is the recent

Durante and Sempi (2015).

An overview paper of the theory and practice with ﬁnancial applications in

mind is given by Embrechts (2009). An important article well known for warning

probabilists, statisticians, and ﬁnancial data analysts alike about fallacies related to

the use of linear correlation to measure dependence is Embrechts et al. (2002). For

a more in-depth introductory treatment of copulas also stressing the latter topics in

the context of quantitative risk management, see McNeil et al. (2015, Chapter 7).

Monographs on copulas mainly targeted towards the realm of ﬁnance are

Cherubini et al. (2004, 2011) and Mai and Scherer (2014); see also Rémillard (2013,

Chapter 8).

An easy-to-digest and statistically oriented introduction to copulas is given by

Genest and Favre (2007); see also Genest et al. (2009) for additional material on

goodness-of-ﬁt testing. A recent statistically oriented monograph is Joe (2014).

A currently active area of research not covered in this book are vine copulas; see

http://vine-copula.org for more details including publications and research projects.

8 1 Introduction

References

Cherubini, U., Luciano, E., & Vecchiato, W. (2004). Copula methods in ﬁnance. Chichester: Wiley.

Cherubini, U., Mulinacci, S., Gobbi, F., & Romagnoli, S. (2011). Dynamic copula methods in

ﬁnance. Chichester: Wiley.

Devroye, L. (1986). Non-uniform random variate generation. New York: Springer.

Durante, F., & Sempi, C. (2010). Copula theory: An introduction. In P. Jaworski, F. Durante,

W. K. Härdle & W. Rychlik (Eds.), Copula theory and its applications (Warsaw, 2009). Lectur e

notes in statistics (pp. 3–32). Berlin: Springer.

Durante, F., & Sempi, C. (2015). Principles of copula theory. Boca Raton, FL: CRC Press.

Embrechts, P. (2009). Copulas: A personal view. Journal of Risk and Insurance, 76, 639–650.

Embrechts, P., & Hofert, M. (2013). A note on generalized inverses. Mathematical Methods of

Operations Research, 77(3), 423–432.

Embrechts, P., McNeil, A. J., & Straumann, D. (2002). Correlation and dependency in risk

management: Properties and pitfalls. In M. Dempster (Ed.), Risk Management: Value at Risk

and Beyond (pp. 176–223). London: Cambridge University Press.

Genest, C., & Favre, A.-C. (2007). Everything you always wanted to know about copula modeling

but were afraid to ask. Journal of Hydrolo gical Engineering, 12, 347–368.

Genest, C., Gendron, M., & Bourdeau-Brien, M. (2009). The advent of copulas in ﬁnance,

European Journal of Finance, 15, 609–618.

Genest, C., Rémillard, B., & Beaudoin, D. (2009). Goodness-of-ﬁt tests for copulas: A review and

a power study. Insurance: Mathematics and Economics, 44, 199–213.

Hoeffding, W. (1940). Massstabinvariante Korrelationstheorie, Schriften des mathematischen

Seminars und des Instituts für Angewandte Mathematik der Universität Berlin, 5, 181–233.

Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.

Joe, H. (2014). Dependence modeling with copulas. Boca Raton, FL: Chapman & Hall/CRC.

Mai, J.-F., & Scherer, M. (2014). Financial engineering with copulas explained. London: Palgrave

Macmillan.

McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts,

techniques and tools (2nd ed.). Princeton, NJ: Princeton University Press.

Nelsen, R. B. (2006). An introduction to copulas. New York: Springer.

Patton, A. J. (2013). Copula methods for forecasting multivariate time series. In G. Elliott

& A. Timmermann (Eds.), Handbook of economic forecasting (pp. 899–960). New York:

Springer.

Rémillard, B. (2013). Statistical methods for ﬁnancial engineering. Boca Baton, FL: Chapman &

Hall/CRC.

Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Extremes in nature: An

approach using copulas. Water science and technology library (Vol. 56). Berlin: Springer.

10 2 Copulas

which is the df of a random vector U = (U

,...,U

) with U

,...,U

ind.

∼ U(0, 1),

that is, with independent U(0, 1)-distributed components; this is easy to see from

the fact that, for any u ∈[0, 1]

, P(U ≤ u) = P(U

≤ u

,...,U

≤ u

) =



j=1

P(U

≤ u

) =



j=1

= (u).

Example 2.1.2 (Independence Copula) In the R package copula, the copula 

is represented by an R object created using the function indepCopula():

To evaluate the copula , the function pCopula() can be used:

To check the numerical equality between computations carried out with

pCopula() on the one hand and directly with (2.1) on the other hand, we can

use the function all.equal(). Also conveniently used in what follows is the

function stopifnot(), which, as the name suggests, produces an error in case at

least one of its arguments evaluates to FALSE:

When d = 2, surface (or perspective) plots and contour (or level) plots of the

function  can be easily produced with wireframe2() and contourplot2(),

respectively, as follows:

The resulting plots are shown in Fig. 2.1. Note that, in this book, we shall use the

expression wireframe plot for a surface or perspective plot by reference to the name

of the corresponding function in the R package lattice of Sarkar (2008). 

Let d = 2 and notice from the wireframe plot in Example 2.1.2 that  is zero

on all edges of the unit square which start at (0, 0),andthat(u

, 1) = u

and

(1,u

) = u

for all u

∈[0, 1]; these two properties are highlighted by dashed

lines in the wireframe plot of Fig. 2.1. In fact, any d-dimensional copula C, d ≥ 2,

is grounded (C(u) = 0ifu

= 0 for at least one j ∈{1,...,d}) and has standard

uniform univariate margins (for any j ∈{1,...,d}, C(1,...,1,u

, 1,...,1) =

for all u

∈[0, 1]). For this reason wireframe2(, FUN = pCopula) by

default visually emphasizes these four line segments for all bivariate copulas. Note

that being grounded and having standard uniform univariate margins alone is not

sufﬁcient for a function C :[0, 1]

→[0, 1] to be a df and thus a copula; for

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