Technische Universität München Fakultät für Mathematik. Properties of extreme-value copulas

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1 Technische Universität München Fakultät für Mathematik Properties of extreme-value copulas Diplomarbeit von Patrick Eschenburg Themenstellerin: Betreuer: Prof. Claudia Czado, Ph.D. Eike Christian Brechmann Abgabetermin: 13. Mai 213

2 Hiermit erkläre ich, dass ich die Diplomarbeit selbständig angefertigt und nur die angegebenen Quellen und Hilfsmittel verwendet habe. München, den 13. Mai 213 i

3 Acknowledgments First of all, I would like to express my sincere gratitude to Prof. Claudia Czado for her excellent supervision of this thesis. I am very thankful for the helpful guidance that I received during the last months. Furthermore, I would like to thank my supervisor Eike Christian Brechmann for his great support, his commitment and many valuable ideas that helped to develop this thesis. Last, but certainly not least, I want to express my deepest gratitude to my parents for their unlimited support and helpful advice throughout my years of study. A deep thank-you also goes to my girlfriend Christina for her patience and encouragement during the last couple of months. ii

4 Contents List of Figures List of Tables List of Abbreviations and Symbols vi vii viii 1 Introduction 1 2 Preliminaries Copulas Tail dependence function Extreme-value copulas Representation of extreme-value copulas Bivariate extreme-value copulas Dependence measures Measures of association Tail dependence Families of bivariate extreme-value copulas Parametric extreme-value copula families Relationships between extreme-value copulas Asymmetry in extreme-value copulas Simulating from extreme-value copulas Asymmetry curve of extreme-value copulas Measure of non-exchangeability Estimation of the measure of non-exchangeability A note on the empirical copula process Generating asymmetric extreme-value copulas Non-parametric tests of symmetry Empirical measure of non-exchangeability A distribution-free test of symmetry Statistical inference on extreme-value copulas Parametric estimation Non-parametric estimation of the Pickands dependence function Estimators based on non-extreme bivariate samples Estimators based on extreme bivariate samples iii

5 iv Contents Moment estimators Weighted minimum distance estimation Corrections of the non-parametric estimators Extreme-value dependence tests Ghoudi, Khoudraji, and Rivest test for extreme-value dependence Extreme-value dependence test based on the copula process Comparison of the extreme-value dependence tests Model selection Graphical model selection Akaike and Bayesian information criteria Goodness-of-fit testing for extreme-value copulas Simulation study Non-parametric estimation of the Pickands dependence function Parametric estimation methods for extreme-value copulas Application Modeling scheme for extreme-value copulas Example Conclusion & Outlook 154 A Additional calculations 156 A.1 Derivatives of bivariate extreme-value copulas A.2 Proof of Theorem A.3 Derivation of the rank-based estimator in (5.73) B Additional figures 162 B.1 Similarity of extreme-value copulas B.2 Dependence measures of extreme-value copulas Bibliography 168

6 List of Figures 2.1 Illustration of an asymmetric copula Illustration of tail dependence Illustration of Example Tail dependence function and stable tail dependence function Pickands dependence function Connection between Pickands- and tail dependence function Illustration of the upper tail dependence coefficient Dependence function of the Marshall-Olkin copula Dependence function, scatter and contour plots of the Hüsler-Reiss copula Dependence function, scatter and contour plots of the t-ev copula Dependence function, scatter and contour plots of the Gumbel copula Dependence function, scatter and contour plots of the Tawn copula Dependence function, scatter and contour plots of the Galambos copula Dependence function, scatter and contour plots of the Joe copula Dependence function, scatter and contour plots of the BB5 copula Relationships between extreme-value copulas Similarity of the Pickands dependence functions of extreme-value copulas Pickands dependence functions with similar dependence characteristics Kendall s τ and λ u of the t-ev copula Relationship between τ and λ u of the t-ev copula Approximation of the t-ev copula Relationship between τ and λ u of the BB5 copula Relationship between τ and λ u of the symmetric Tawn copula Relationship between τ and λ u of the symmetric Joe copula Illustration of symmetry for the Gumbel copula Asymmetry curve of extreme-value copulas Illustration of Table Maximally non-exchangeable extreme-value copulas Estimation of the measure of non-exchangeability Power of the non-parametric symmetry test for n = 5 observations Power of the non-parametric symmetry test for n = 1 observations Power of the non-parametric symmetry test for n = 2 observations Theoretical functions h for the symmetry test Greatest convex minorant of non-parametric estimators v

7 vi List of Figures 6.1 Mean integrated squared errors of the non-parametric estimators Variation in parametric estimation methods for one-parameter families Variation in parametric estimation methods for two-parameter families Variation in parametric estimation methods for three-parameter families Modeling scheme for extreme-value copulas Scatter- and empirical contour plot of the pairs (Cs,Sc) and (Cs, Sc) Empirical density estimate of t for the pair (Cs, Sc) Fit of the estimated Pickands dependence functions Fit of the estimated Pickands dependence functions Scatter plot of the pair (Cs, Sc) with superimposed asymmetry curve. 152 B.1 Similarity of extreme-value copulas for τ = B.2 Similarity of extreme-value copulas for τ = B.3 Dependence measures of the Hüsler-Reiss copula B.4 Dependence measures of the t-ev copula B.5 Dependence measures of the Gumbel copula B.6 Dependence measures of the Tawn copula with θ = B.7 Dependence measures of the Tawn copula with θ = B.8 Dependence measures of the Galambos copula B.9 Dependence measures of the BB5 copula B.1 Dependence measures of the Joe copula with δ = B.11 Dependence measures of the Joe copula with δ =

8 List of Tables 3.1 Dependence coefficients of the families in Figure Pickands dependence functions of extreme-value copula families Approximation of the t-ev copula Approximation of mode{h(t)} Simulation study for empirical version of measure of non-exchangeability Parameter sets for the non-parametric symmetry test Rejection rates for the symmetry test for n = 5 observations Rejection rates for the symmetry test for n = 1 observations Rejection rates for the symmetry test for n = 2 observations Overview of non-parametric estimators of the Pickands dependence function Non-parametric estimators of the Pickands dependence function Overview of extreme-value dependence tests Mean integrated squared error of the non-parametric estimators Inversion of Blomqvist s β Comparison of parametric estimation methods P-values of the extreme-value dependence test for the pair (Cs, Sc) Estimated measures of the pair (Cs, Sc) Parametric estimation of the pair (Cs, Sc) vii

9 List of Abbreviations and Symbols BIPIT DA R Z Γ( ) Bivariate probability integral transform Domain of attraction Real numbers Integral numbers Gamma function X Random variable x Observation/realization of the random variable X x Sample mean, mean of the n observations of the random variable X n Sample size, number of observations of the random variable X (Û1, Û2) Vector of pseudo-observations X = (X 1,..., X d ) Vector of random variables x = (x 1,..., x d ) Vector of observations of the random variables X 1,..., X d d Dimension, number of variables F ( ) f( ) F 1 ( ),..., F d ( ) f 1 ( ),..., f d ( ) F ( ) Distribution function Density function Marginal distribution functions Marginal density functions Conditional distribution function E[ ] Expectation Var( ) Variance Cov(, ) Covariance Corr(, ) Correlation 1( ) Indicator function U[a, b] Uniform distribution on [a, b] Φ ρ ( ) Cdf of the standard normal distribution with correlation coefficient ρ Φ ( ) Inverse of the standard normal distribution function T ν ( ) Cdf of Student s t distribution with ν degrees of freedom C( ) Ĉ( ) C( ) Joint survival function Survival copula distribution function Copula distribution function viii

10 List of Abbreviations and Symbols ix c( ) C n ( ) C n C φ ( ) C κ1,κ 2 ( ) C φ,a ( ) Π d ( ) θ l( ) l ( ) l( ) K( ) A( ) Â n ( ) Â n,c ( ) Copula density function Empirical copula Empirical copula process Archimedean copula distribution function Copula generated by the device of Khoudraji Archimax copula distribution function d-dimensional independence copula Parameter or parameter vector of the copula C( ) Upper tail dependence function Stable tail dependence function Lower tail dependence function Kendall s distribution function Pickands dependence function Rank-based estimator of Pickands dependence function Corrected version of rank-based estimator of Pickands dependence function ˆρ n ρ S ˆτ τ β ˆβ n ˆλ u n λ u ˆλ l n λ l EV Empirical version of Spearman s ρ S Spearman s ρ Empirical version of Kendall s Tau Kendall s Tau Blomqvist s β Empirical version of Blomqvist s β Empirical version of upper tail dependence coefficient Upper tail dependence coefficient Empirical version of lower tail dependence coefficient Lower tail dependence coefficient Class of bivariate extreme-value copulas

11 Chapter 1 Introduction While the modeling of the dependence structure of random variables is one of the main topics in probability theory and statistics, the dependencies among extreme events have been of special interest lately, fuelled by events like the financial crisis or natural catastrophes. It was already pointed out by Coles and Tawn [1991] that most extreme events are in character multivariate. This means that extreme events are characterized by the joint occurrence of marginal extremes. As an example consider the financial sector, where substantial risks for companies arise if several extremal losses happen simultaneously. Consequently, the dependence of extremes has to play an important role in the modeling of joint extremes. In the classical approach of multivariate extreme-value theory, joint extremes are modeled by the asymptotic limit of the joint distribution of componentwise maxima of independent and identically distributed (i.i.d.) samples (Resnick [1987], Galambos [1987]). Joe [1997] investigates possible dependence structures of the limiting distributions, but a tremendous gain in flexibility is achieved if one uses copulas to model the dependence. Copulas provide a functional link between multivariate distribution functions and their univariate margins (Sklar [1959]). This allows to model the underlying dependence structure separately from the margins, a result which explains the growing popularity of these functions (see Nelsen [26] for a monograph on copulas). Extreme-value copulas combine the areas of multivariate extreme-value theory and copulas. They arise as asymptotic limits of copulas of component-wise maxima in i.i.d. samples and, thus, can be considered as appropriate models for the dependence structure between extreme events. However, compared to the research on copulas, the study of extreme-value copulas is quite behind. Recently, Gudendorf and Segers [21] have conducted a survey which grasps the most important aspects of extreme-value copulas and provides a bibliography for further literature. The aim of this thesis is to gather the literature available on extreme-value copulas and investigate their properties in detail to gain a comprehensive picture of this copula class. Therefore we will put special focus on statistical methods that are particularly related to the extreme-value copula class. The thesis is organized as follows. In Chapter 2, we will lay the theoretical foundation for this thesis. Basic concepts of copulas will be discussed. As mentioned, most results for modeling joint extremes have been derived in the context of multivariate extremes. We will combine these results with the concept of copulas and derive the representation of extreme-value copulas step by step. In the bivariate case, extreme- 1

12 2 Chapter 1 Introduction value copulas can be uniquely characterized by a one-dimensional mapping, called the Pickands dependence function. In Chapter 3, the most popular extreme-value copulas will be introduced and illustrated by means of this function. Further, connections between the different copula families will be highlighted. In contrast to some other copula classes (e.g. the popular Archimedean copula class), extreme-value copulas are capable of modeling asymmetry. In Chapter 4 we will discuss this aspect in detail and present a construction principle to generate asymmetric extreme-value copulas. Chapter 5 is concerned with statistical inference on extreme-value copulas. Besides estimation methods, tests for the hypothesis of an extreme-value dependence structure will be introduced, which should be considered before using this copula class. In Chapter 6 the estimation methods are compared in a Monte Carlo study. Chapter 7 discusses extreme-value copulas from a practical perspective. A summarizing modeling scheme for extreme-value copulas will be presented and illustrated by means of an example. Chapter 8 concludes, points out recent fields of research and potential fields of improvement for the future.

13 Chapter 2 Preliminaries This chapter provides the stochastic framework and basic definitions for the thesis. A sound understanding of the underlying theory is a crucial prerequisite before working with the matter. Therefore we will first build up the theory of extreme-value copulas in three constructive steps: after introducing the concept of copulas as the state-of-the-art methodology in dependence modeling, we will present tail dependence functions as an important tool, and finally give the common representation for the class of extremevalue copulas. 2.1 Copulas According to Genest and Favre [27], a main limitation of the traditional approach to multivariate dependence modeling is the fact that the individual behaviour of the margins has to be described by a single parametric family. As will become clear in the following section, working with copulas overcomes this limitation by splitting the analysis of the joint distribution of a multivariate random vector into two parts: the analysis of the margins - which is carried out independently for each random variable - and the choice of a copula which describes the dependence structure as a function of the margins. The copula thus provides a link (latin word: copula) between univariate margins and the joint distribution and implicitly defines a multivariate distribution function. Further it allows a margin-free analysis of multivariate dependence structures. We will first give a formal definition of copulas and then discuss some important properties. Main references for this chapter are Nelsen [26], Embrechts et al. [23] and Joe [1997]. As most of our work will be performed in two dimensions, we first define the bivariate copula. An extension to the higher dimensional case works in analogous fashion, however requires a little more theoretical background regarding multivariate distribution functions for which we refer to the given references. Definition 2.1 (Bivariate Copula) A bivariate (or two-dimensional) copula is a function C : [, 1] 2 [, 1] with the following properties: (i) For every u,v in [, 1] C(u, ) = = C(, v) 3

14 4 Chapter 2 Preliminaries and C(u, 1) = u and C(1, v) = v. (ii) C is 2-increasing, i.e. for every u 1, u 2, v 1, v 2 in [,1] s.t. u 1 u 2 and v 1 v 2, C(u 2, v 2 ) C(u 2, v 1 ) C(u 1, v 2 ) + C(u 1, v 1 ). Properties (i) and (ii) define a bivariate copula as a distribution function on [, 1] 2 with uniformly distributed margins (Nelsen [26], Embrechts et al. [23]). Thus a bivariate copula defines a bivariate distribution function on [, 1] 2 with uniformly distributed margins, i.e. for random variables U 1, U 2 C it holds that U 1 U[, 1] and U 2 U[, 1]. If the matter of interest is the survival probability of U 1, U 2 beyond points u 1, u 2, i.e. P (U 1 > u 1, U 2 > u 2 ), we have the following definition. Definition 2.2 (Joint survival function) Let U := (U 1, U 2 ) be a random vector with copula C. Then the joint survival function C of U is defined as C(u 1, u 2 ) = P (U 1 > u 1, U 2 > u 2 ). Note that this is not a copula. However there exists a survival copula Ĉ and the connection between the survival function C and the survival copula Ĉ is given by C(u 1, u 2 ) = Ĉ(1 u 1, 1 u 2 ). The ideas mentioned so far can be transferred to the multidimensional case as follows. Definition 2.3 (Copula) A d-dimensional copula is a function C : [, 1] d [, 1] with the following properties: (i) For every u := (u 1,..., u d ) [, 1] d and C(u) = if at least one coordinate of u is, if all coordinates of u are 1 except u k, k {1,..., d}, then C(u) = u k. (ii) C is d-increasing (for details see Nelsen [26], Embrechts et al. [23]). Besides the fact that a copula is a multivariate distribution function, the popularity in applications is mostly due to Sklar s Theorem (Sklar [1959]). This central theorem for copulas provides the link between joint distribution functions and their univariate margins and hence builds the foundation for most copula applications. Theorem 2.4 (Sklar) Let F be a d-dimensional distribution function with margins F 1,..., F d.then there exists a d-dimensional copula C such that for all x = (x 1,..., x d ) (R {, }) d : F (x 1,..., x d ) = C(F 1 (x 1 ),..., F d (x d )). If F 1,..., F d are all continuous, then the copula C is unique.

15 2.1 Copulas 5 The theorem illustrates the separation of a multivariate distribution function into univariate margins and the copula C, with C modeling the dependence structure. We will always consider continuous marginals in this thesis and hence assume the uniqueness of the respective copula. Precisely, consider the case of a random vector X := (X 1,..., X d ) with continuous margins F 1,..., F d and joint distribution function F. Then the distribution of the random vector X can be written as F (x 1,..., x d ) = P (X 1 x 1,..., X d x d ) = C(F 1 (x 1 ),..., F d (x d )). We say that C is the unique copula of X. Note that the margins of a copula have to be uniform. This is achieved by the probability integral transformation X i F i (X i ), formally defined below. Theorem 2.5 (Probability integral transform) Let X be a random variable with continuous distribution F. Then the random variable U = F(X) is uniformly distributed on (,1). Proof. F is a distribution function and thus non-decreasing. Denote by F : (, 1) (, ), F (u) := inf{x : F (x) u} the generalised inverse of F. Then P (U u) = P (F (X) u) = P (X F (u)) = F (F (u)) = u, < u < 1, which is the distribution function of a uniformly distributed random variable on (,1), so that U U(, 1). As a copula is a distribution function, it is possible to derive the copula density c which is needed in many applications, e.g. in maximum likelihood estimation. For a d-dimensional copula C it is defined as c(u 1,..., u d ) = d C(u 1,..., u d ) u 1... u d, (2.1) given that all partial derivatives exist. It follows from the above definition that if C is the unique copula of a d-dimensional random vector X F with continuous margins F i, i = 1,..., d, the probability density f of X is given by d f(x) = c(f 1 (x 1 ),..., F d (x d )) f i (x i ). So just as the joint distribution function, the joint density function can be separated into the marginal densities and the copula density. This motivates the following theorem. We will also give a proof of this theorem to illustrate some aspects of the theory presented so far. Theorem 2.6 Let X := (X 1,..., X d ) be a vector of continuous random variables. Then X 1,..., X d are independent if and only if C is the independence copula Π d (u) = u 1... u d. i=1

16 6 Chapter 2 Preliminaries Proof. We will carry out the proof in two steps. First, it has to be shown that the independence copula is indeed a copula. We will prove this in the bivariate case. Second, the statement itself will be proven. (1) For the independence copula Π 2 to be a bivariate copula, the requirements of Definition 2.1 have to be fulfilled. Property (i) follows directly from the definition of the independence copula, property (ii) can be calculated as Π 2 (u 2, v 2 ) Π 2 (u 2, v 1 ) Π 2 (u 1, v 2 )+ Π 2 (u 1, v 1 ) = u 2 v 2 u 2 v 1 u 1 v 2 +u 1 v 1 = u 2 (v 2 v 1 ) u 1 (v 2 v 1 ) = (u 2 u 1 )(v 2 v 1 ), where the last inequality follows from the assumptions u 2 u 1 and v 2 v 1. The same results hold in the multivariate case. (2) Assume that the X i are independent. Then F (x) = F 1 (x 1 )... F d (x d ). From Sklar (1959) we know that F can be expressed via a copula C as F (x) = C(F 1 (x 1 ),..., F d (x d )). Thus C(F 1 (x 1 ),..., F d (x d )) = F 1 (x 1 )... F d (x d ) C = Π d. The uniqueness of the solution is guaranteed by the continuity of the margins. C(u) = Π d (u) c(u) = π d (u) = d Π d (u) d = 1 f(x) = 1 f i (x i ) X i u 1... u d i=1 independent for i 1,..., d We now state a couple of important properties of copulas. Theorem 2.7 (Fréchet-Hoeffding bounds) Consider the following functions defined on the unit cube [, 1] d : M d (u) = min{u 1,..., u d }, (2.2) W d (u) = max{u u d d + 1, }. Then for any u [, 1] d and any d-dimensional copula C it holds that W d (u) C(u) M d (u). The function M d is a copula in any dimension d. In contrast W d is not a copula for d 3, but it still provides the best available lower boundary for C in any dimension (Nelsen [26], Theorem ). The functions M and W also have descriptive interpretations, as the copula M describes perfect co-monotonicity between the random variables while W corresponds to the case of counter-monotonicity. Together with the independence copula Π the stated functions cover the three extreme scenarios of possible dependence structures. Theorem 2.8 Let (X 1,..., X d ) be a vector of continuous random variables with copula C. If α 1,..., α d are strictly increasing functions, then (α 1 (X 1 ),..., α d (X d )) also has copula C. The fact that copulas are invariant under strictly increasing transformations of the random variables provides a useful tool in applications and further explains the popularity of copulas in multivariate dependence modelling. A question that will concern us in this thesis is the matter of symmetry of copulas. We will give a short definition here (Durante and Papini [29]), an exhaustive treatment of the topic will take place in Chapter 4. An illustration is given in Figure 2.1.

17 2.2 Tail dependence function U1 U U1 U2 Figure 2.1: Scatter plots of a symmetric copula (left) and an asymmetric copula (right). Definition 2.9 (Symmetric copula) Let X, Y be continuous random variables with copula C. X and Y are called exchangeable if and only if they are identically distributed, i.e. F X = F Y, and the copula C is symmetric, i.e. C(u, v) = C(v, u). 2.2 Tail dependence function Tail dependence is a measure of comovement in the tails of multivariate distribution functions, i.e. of joint extreme values of random variables. Informally spoken, it gives the probability of large (small) values of one random variable co-occuring with large (small) values of other random variables. The relevance to extreme value theory is thus obvious. In the following we will introduce the concept of tail dependence functions of Joe et al. [21]. Figure 2.2 gives a graphical motivation for the topic in the two-dimensional case. Let (U 1, U 2 ) C, where C is a bivariate copula. Scatter plots of two possible copula models are displayed in Figure 2.2. We are interested in comovements of extremely large (small) values, i.e. points in the far right upper (left lower) corner. The points in the boxes indicate that the left plot displays tail independence as there is no connection between large (or small) values of the respective variables while in the right plot, especially in the upper right corner, there is an obvious coexistence of large values of both variables. For a formal definition, let F be the distribution function of some random vector X = (X 1,..., X d ) with copula C. In analogy to Definition 2.2, where the bivariate case was considered, we denote by C the joint survival function of C which is given by C(u 1,..., u d ) = P (U 1 > u 1,..., U d > u d ) =: Ĉ(1 u 1,..., 1 u d ). (2.3)

18 8 Chapter 2 Preliminaries U1 U U1 U2 Figure 2.2: Scatter plots of the independence copula (left) and a copula displaying upper tail dependence (right). The function Ĉ is called the survival copula of C. The probability of joint occurrence of extreme values is then given by the tail dependence functions. Definition 2.1 (Tail dependence functions) The lower and upper tail dependence functions denoted by l and l, respectively, are defined as l(w, C) = lim s s 1 C(sw 1,..., sw d ) w = (w 1,..., w d ) R d + (2.4) l(w, C) = lim s s 1 C(1 sw 1,..., 1 sw d ) w = (w 1,..., w d ) R d + (2.5) Any result for one of the tail dependence functions can be transferred to the other one by (2.3), from which it follows that Ĉ(uw 1,..., uw d ) = P (1 U 1 u 1,..., 1 U d U d ) = C(1 uw 1,..., 1 uw d ) and as a consequence l(w, Ĉ) = l(w, C). Thus in the following we will focus on l(w, C) as upper tail dependence functions are of special relevance to extreme-value copulas. For notational ease we will simply speak of the tail dependence function and denote it by l whenever no risk of confusion arises. Proposition 2.11 (Properties of tail dependence functions) (l1) l is grounded, i.e. l(w) = if at least one w i =, i {1,..., d}. (l1) l is d-increasing, i.e. l(w 1,..., w i + ε i,..., w d ) l(w 1,..., w i,..., w d ) ε i, 1 i d. (l1) l is homogeneous of order one, i.e. l(sw) = sl(w) for any s.

19 2.2 Tail dependence function 9 (l1) l(w) = for all w if and only if l(k) = for some positive k = (k 1,..., k d ). A proof can be found in Joe et al. [21]. Note that properties (l1) and (l2) are very similar to those of copulas introduced in Section 2.1. We now turn our attention to an important aspect of Definition 2.1. The vector w defines the relative scale at which the components of the copula C go to extremes. This is best explained in an example. Example 1 Consider the bivariate case, i.e. w R 2 +. (1) Choose w 1 = w 2 =.5. Then l(.5,.5) = lim s s 1 C(s.5, s.5) = lim s s 1 P (U 1 1.5s, U 2 1.5s), i.e. the tail dependence function gives the probability that U 1 and U 2 take on extremely high values beyond a threshold lim(1.5s), while the threshold values s are identical. (2) Now choose w 1 w 2, for instance w 1 =.8, w 2 =.2. Then l(.8,.2) = lim s s 1 C(s.8, s.2) = lim s s 1 P (U 1 1.8s, U 2 1.2s). In this case the tail dependence function gives the joint probability of U 1 and U 2 exceeding different thresholds, since U 1 may take on lower values than U 2. This is illustrated in Figure 2.3. While technically the definition holds for s, s was first fixed at the values.4,.2 and.1 for better illustration (s means that the boxes move further into the upper right corner). For fixed s, the edges of the boxes correspond to the threshold values of U 1 and U 2 in the tail dependence function. The left plot shows case (1) where the boxes are symmetric as both threshold values are identical. In case (2), U 1 may take on smaller values than U 2, which is reflected in the asymmetric boxes in the right plot including a broader range of U 1 values. As s is continuous, repeating this procedure for a wide range of s values, here s [,.6], results in the lines through the corners of the boxes which display the threshold values for U 1 and U 2 for the respective values of s. Case (2) of Example 1 can be interpreted as measuring dependence along an axis other than the diagonal. This is an important extension which will be considered again in Chapter 4 during the investigation of asymmetry.

20 1 Chapter 2 Preliminaries U1 U U1 U2 Figure 2.3: Illustration of Example 1 for w = (.5,.5) (left) and w = (.8,.2) (right). 2.3 Extreme-value copulas Representation of extreme-value copulas In this section we will derive the most common representation of extreme-value copulas which is given via the so-called Pickands dependence function. Due to the results in Section 2.1, copulas are a growingly popular way of expressing classical distribution functions via their margins. However, the main theoretical results for this chapter were already provided by Haan and Resnick [1977] and Pickands [1981] in the context of multivariate extremes. They show that the class of extreme-value distributions coincides with the class of max-stable distributions and give a handy representation of this class using the tail dependence function. A transfer of this theory to the concept of copulas is given by Gudendorf and Segers [21], which we will mainly follow and complement here. Consider a sample of n independent d-dimensional random vectors X (i) := (X i1,..., X id ), i {1,..., n} with joint marginal distribution F, marginals F i and copula C F. Being interested in extremes we denote the vector of component-wise maxima by M n = (M n1,..., M nd ), where M nj = max i=1,...,n {X ij}. Note that M n will normally not be a sample point, as no observation takes on the extremes in all margins simultaneously. However, this is the vector that was historically considered for the investigation of multivariate extremes and many important statistical tools can be derived from the study of this vector (Beirlant et al. [24]).

21 2.3 Extreme-value copulas 11 Proposition 2.12 Let C F be the copula of the random vector X = (X 1,..., X d ). Then the copula C Mn of M n is given by C Mn (u 1,..., u d ) = C F (u 1/n 1,..., u 1/n d ) n, (u 1,..., u d ) [, 1] d. (2.6) Proof. First we show that the margins F n,i of M n are given by F n i : F ni (x i ) = P (M ni x i ) = P (X 1i x i,..., X di x i ) = P (X 1i x i )... P (X di x i ) = F i (x i ) n, where we use the independence of the sample. Thus we see that for u i = F ni (x i ) C Mn (u 1,..., u d ) = C Mn (F n1 (x 1 ),..., F nd (x d ) = P (M n1 x 1,..., M nd x d ) = P (X 11 x 1,..., X n1 x 1,..., X 1d x d,..., X nd x d ) = P (X 11 x 1,..., X 1d x d,..., X n1 x 1,..., X nd x d ) iid = P (X 11 x 1,..., X d1 x d ) n = C F (F 1 (x 1 ),..., F d (x d )) n = C F (F n,1 (x 1 ) 1/n,..., F nd (x d ) 1/n ) n = C F (u 1/n 1,..., u 1/n d ) n. We are interested in the asymptotic distribution of {M n }, that means we have to check if the copula C Mn converges to some copula C. If so, this limiting copula is called an extreme-value copulas. Definition 2.13 (Extreme-value copula) A copula C is called an extreme-value copula if there exists a copula C F such that, for n, C F (u 1/n 1,..., u 1/n d ) n C(u 1,..., u d ) (u 1,..., u d ) [, 1] d (2.7) The copula C F is said to be in the domain of attraction of the copula C (written C F DA(C) ). This definition formally introduces extreme-value copulas as asymptotic limits of componentwise maxima, however it does not provide a handy statistical tool. In order to be able to work with this class of copulas, a different representation is sought. By definition, extreme-value copulas are the copulas of extreme-value distributions. A major setback of multivariate extreme-value distributions in dimension d 2, however, is that the class of possible limiting dependence structures is infinite and therefore cannot be captured in a finite-dimensional parametric family (Beirlant et al. [24]). Thus unlike the univariate case, where the GEV (Generalised Extreme-Value) distribution covers the whole range of possible limit functions and provides a handsome tool with just three parameters, multivariate extreme-value distributions have to be indexed in another way. The most popular approach is closely linked to the concept of max-stability.

22 12 Chapter 2 Preliminaries Definition 2.14 (Max-stability of copulas) A copula C is called max-stable if it satisfies the relationship C(u 1,..., u d ) = C(u 1/m 1,..., u 1/m for every integer m 1 and (u 1,..., u d ) [, 1] d. This definition gives rise to the following theorem. d ) m Theorem 2.15 A copula C is an extreme-value copula if and only if it is max-stable. Proof. Let C Mn denote the copula of M n. We first rewrite C n as C Mn (u 1,..., u d ) (2.6) = C F (u 1/n 1,..., u 1/n d ) n ( ) = C F (u 1/mk 1,..., u 1/mk d ) mk = C F ((u 1/m 1 ) 1/k,..., (u 1/m ) 1/k ) mk d (2.6) = C Mk (u 1/m 1,..., u 1/m d ) m, (2.8) where in ( ) we used the substitution n = mk for fixed integer m and we let k in the last step. Now using the fact that C is an extreme-value copula it follows from Definition 2.13 that C(u 1,..., u d ) (2.7) = lim C F (u 1/n 1,..., u 1/n n d (2.6) = lim n C Mn (u 1,..., u d ) ) n (2.8) = lim C Mk (u 1/m 1,..., u 1/m d k = C(u 1/m 1,..., u 1/m d ) m. This follows directly from the definition of max-stability. Knowing the class of extreme-value distributions, we are interested in an applicable representation. Therefore we need the following definition. ) m Definition 2.16 (Max-id and max-stable distribution function) A distribution function F on R d is max-infinitely divisible (max-id) if for every t there exists a distribution F t on R d such that F = F t t, i.e. F 1/t is a distribution function. max-stable, if for every t > there exist functions a i (t) >, b i (t), i = 1,..., d such that F t (x) = F (a 1 (t)x 1 + b 1 (t),..., a d (t)x d + b d (t)),. (2.9)

23 2.3 Extreme-value copulas 13 For every max-id distribution function F it is further known that there exists a measure µ such that F (x) = exp( µ([, )\[, x)), x [, ], (2.1) (see Resnick [1987], Proposition 5.8). Due to this representation the measure µ is called exponent measure. It is clear from (2.9) that if F is max-stable, F t is a distribution function for every t > and therefore every max-stable distribution is also max-id. Thus every max-stable distribution can also be represented via an exponent measure. By definition, the set of copulas of max-stable distributions is exactly the class of maxstable copulas, which coincides with the class of extreme-value copulas according to Theorem We will see in the following theorem, which is found in Gudendorf and Segers [21], that the representation of extreme-value copulas is closely related to this representation via an exponent measure. Theorem 2.17 A d-variate copula C is an extreme-value copula if and only if there exists a finite Borel measure H on the unit simplex d 1 = {(u 1,..., u d ) [, ) d : j u j = 1}, called spectral measure, such that C(u 1,..., u d ) = exp{ l ( log u 1,..., log u d )}, (u 1,..., u d ) (, 1] d, (2.11) where the stable tail dependence function l : [, ) d [, ) is given by l (x 1,..., x d ) = max {w jx j }dh(w 1,..., w d ), (x 1,..., x d ) [, ) d. j=1,...,d d 1 The spectral measure H is arbitrary except for the d moment constraints w j dh(w 1,..., w d ) = 1, j {1,..., d}. d 1 Proof. The proof of this theorem is mainly based on the representation of max-stable distributions via spectral measures and requires transformation to polar coordinates. It can be found, for instance, in Resnick [1987] or Beirlant et al. [24]. To provide a more intuitive interpretation of the stable tail dependence function 1 other than via the spectral measure H, we use the following Proposition. Proposition 2.18 The domain of attraction equation (2.7) is equivalent to lim t t 1 (1 C F (1 tx 1,..., 1 tx d ) = l (x 1,..., x d ). 1 Note that the nomenclature of the terms tail dependence function and stable tail dependence function is very diverse in the literature, even in the sources cited in this section. For instance, Beirlant et al. [24] defines a different tail dependence function in a way that the stable tail dependence function arises as its limit for values in the far corners. In Joe et al. [21], the stable tail dependence function is called exponent function, expressing the use in the representation of extreme-value copulas via an exponent measure. In other articles, l is also referred to as upper tail copula (Haug et al. [211],Schmidt and Stadtmüller [26]).

24 14 Chapter 2 Preliminaries Proof. In (2.7), we rewrite the extreme-value copula C on the right-hand side in terms of its stable tail dependence function as in (2.11). This yields lim C F (u 1/n 1,..., u 1/n n d ) n = exp{ l ( log u 1,..., log u d )} lim log(c F (u 1/n 1,..., u 1/n n d ) n ) = l ( log u 1,..., log u d ) (1) lim n n(1 C F (exp{ 1 n log u 1},..., exp{ 1 n log u d})) = l ( log u 1,..., log u d ) (2) lim n n(1 C F (1 + 1 n log u 1,..., n log u d)) = l ( log u 1,..., log u d ) lim t t 1 (1 C F (1 tx 1,..., 1 tx d )) = l (x 1,..., x d ) (2.12) with the substitutions x i = log u i, i {1,..., d} and t = 1 in the last step. Further n we used the linear expansions of the logarithm and the exponential function in (1) : log(x) 1 x for x and in (2) : exp( 1 x) x for n. n n As 1 C F (1 u 1,..., 1 u d ) = P ( d i=1 {U i > 1 u i }), the stable tail dependence function l (x 1,..., x d ) is proportional to the limit lim P ( d i=1 {U i > 1 tx i }). Precisely, t recall from (2.12) that ( d ) l (x 1,..., x d ) = lim t 1 (1 C F (1 tx 1,..., 1 tx d )) = lim t 1 P {U i > 1 tx i }. t t i=1 (2.13) In contrast to that, the tail dependence function l as introduced in (2.5) is defined as l(x 1,..., x d ) = lim t t 1 C F (tx 1,..., tx d ) = lim t ( d ) t 1 P {U i > 1 tx i }, i=1 which expresses joint exceedance of the thresholds of all margins U i. This is graphically illustrated in Figure 2.4. A connection between the two functions is given by the inclusion-exclusion-principle and formulated in Joe et al. [21]: l (x 1,..., x d ) = ( 1) S 1 l(x 1,..., x j ; j S; C S ) S I,S for all non-empty subsets S I, where I := {1,..., d} denotes the index set and C S the copula of the S -dimensional margin {U i, i S}. For S = 1, define l(x i ) = x i. As an example, in the bivariate case this leads to the important relation l (x 1, x 2 ) = x 1 + x 2 l(x 1, x 2 ). (2.14) In the following we will state two important properties of the stable tail dependence function.

25 2.3 Extreme-value copulas U1 U U1 U2 Figure 2.4: Left panel: The stable tail dependence function l is proportional to the probability that at least one component is above a certain threshold. Right panel: For the tail dependence function l both components have to be large simultaneously. Proposition 2.19 The stable tail dependence function l as in (2.12) has the following properties. (i) l is homogeneous of order 1, i.e. l (ax) = al (x), a >, (x 1,..., x d ) [, ) d. (2.15) (ii) max{x 1,..., x d } l (x 1,..., x d ) x x d. Proof. (i) l (ax) (2.12) = lim t t 1 (1 C F (1 atx 1,..., 1 atx d )) = lim n n{1 C F (1 ax 1 n,..., 1 ax d n )} = lim s as{1 C F (1 ax 1 as,..., 1 ax d as )} = lim s as{1 C F (1 x 1 s,..., 1 x d s )} = lim t at 1 (1 C F (1 tx 1,..., 1 tx d )) (2.12) = al (x). (ii) From (2.13) it is known that l is proportional to the union of the probabilities P (U i > 1 tx i ), i = 1,..., d. As U U[, 1], each of these probabilities equals P (U i > 1 tx i ) = 1 P (U i 1 tx i ) = tx i.

26 16 Chapter 2 Preliminaries Thus we can establish the following boundaries: ( d ) max{tx 1,..., tx d } P {U i > 1 tx i } tx tx d. (2.16) i=1 Multiplying (2.16) by 1/t and letting t gives max{x 1,..., x d } l (x 1,..., x d ) x x d, (2.17) see also (2.13). The homogeneity property (i) in Proposition (2.19) allows to restrict the stable tail dependence function to the unit simplex d 1 := {(w 1,..., w d ) [, 1] d : w w d = 1}. This idea is originally suggested in Pickands [1981]. Therefore consider the real a := 1 x x d as a scaling factor in (2.15). The vector x := ax is then a normed version of x as it holds that x i := ax i = x i x x d [, 1], i = 1,..., d Thus one can define a function on the unit simplex d 1 by and d x i = 1. A( x) := l (ax) = al (x). (2.18) The function A(x) is the restriction of the stable tail dependence function to the unit simplex. That idea was originally formulated in Pickands [1981], which motivates the following definition. Definition 2.2 (Pickands dependence function) The Pickands dependence function A : d 1 [1/d, 1] is defined as A(w 1,..., w d ) := l (x 1,..., x d ) d i=1 x, where w j := i for (x 1,..., x d ) [, ) d \{}. i=1 x j d i=1 x, j = 1,..., d, (2.19) i Note that the variable w d = 1 w 1... w d 1 is often suppressed and A is written only as a function of (w 1,..., w d 1 ). The fact that the Pickands dependence function maps to [1/d, 1] follows from using (2.17) and (2.19). According to (2.17) it holds that max{x 1,..., x d } l (x 1,..., x d ) x x d max{x 1,..., x d } l (x 1,..., x d ) x x d x x }{{ d } A(w 1,...,w d 1 ) x x d x x d (2.2)

27 2.3 Extreme-value copulas 17 In case that all x i, i = 1..., d, are identical, the left-hand side in (2.2) equals 1/d. Otherwise it holds that the fraction is larger than 1/d, from which the claim follows. Definition 2.2 captures the connection between the Pickands dependence function and the stable tail dependence function. Together with (2.11) this gives rise to the most common representation of extreme-value copulas which are expressed via A as {( d ) ( C(u 1,..., u d ) = exp log u i A i=1 log u 1 d i=1 log u,..., i log u d 1 d i=1 log u i Proposition 2.21 (Properties of the Pickands dependence function) (A1) A is convex. (A2) max{w 1,..., w d 1 } A(w 1,..., w d 1 ) 1 (w 1,..., w d 1 ) d 1. )}. (2.21) Proof. A proof of Lemma 2.21 can be found in Joe [1997] (p.175). It is shown that (A2) has to hold in order for C in (2.21) to define a copula. Further it follows that in this case A, if it exists, is positive definite and therewith convex. In the next section we will discuss these properties in more detail in the important bivariate case Bivariate extreme-value copulas Most of the literature available on extreme-value copulas concentrates on the bivariate case. This has mainly two reasons. First, higher dimensional copulas of dimension d 3 have certain limitations (e.g. limited number of available families, lack of flexibility) and are therefore often modeled by pair-copula constructions built up by bivariate copulas (Aas et al. [29]). Second, for bivariate extreme-value copulas, the Pickands dependence function reduces to a one-dimensional mapping which completely characterizes C and facilitates many statistical procedures. In the bivariate case, the stable tail dependence function can be restricted to the unit simplex, which we parameterise following Gudendorf and Segers [21]: 1 = {(1 t, t) : t [, 1]}. (2.22) We know from Theorem 2.17 (precisely (2.11)) that we can express a bivariate extremevalue copula as C(u 1, u 2 ) = exp{ l ( log(u 1 ), log(u 2 ))}. Defining w 1 := 1 t and w 2 := t, according to (2.19) the Pickands dependence function is then defined as A(1 t, t) = l ( log(u 1 ), log(u 2 )) log(u 1 ) log(u 2 ) homog. = l (log(u 1 ), log(u 2 )) (log(u 1 ) + log(u 2 )) homog. = l ( log(u1 ) log(u 1 u 2 ), log(u 2 ) log(u 1 u 2 ) ). (2.23)

28 18 Chapter 2 Preliminaries As mentioned by Segers [212b], it is common to suppress one argument of the Pickands dependence function due to the restriction on the unit simplex. As it is more convenient to work with A(t) than A(1 t), we will work with where A(t) (2.23) = l ( log(u1 ) log(u 1 u 2 ), log(u 2 ) log(u 1 u 2 ) ) = l (1 t, t), (2.24) t = log(u 2) log(u 1 u 2 ). (2.25) We will now define bivariate extreme-copulas. Theorem 2.22 (Bivariate extreme-value copula) A bivariate copula C is an extreme-value copula if and only if { ( )} log(u2 ) C(u 1, u 2 ) = exp (log(u 1 ) + log(u 2 ))A log(u 1 u 2 ) = (u 1 u 2 ) A(log(u 2)/ log(u 1 u 2 )), (u 1, u 2 ) (, 1] 2 \{(1, 1)}, (2.26) where A : [, 1] [1/2, 1] is convex and satisfies max{1 t, t} A(t) 1 t [, 1]. Proof. Multivariate extreme-value copulas are defined in (2.11), which in the bivariate case reduces to C(u 1, u 2 ) = exp{ l ( log(u 1 ), log(u 2 ))}. (2.27) Using (2.23) we obtain (2.28) in (2.27) yields A(t) = l ( log(u 1 ), log(u 2 )) log(u 1 ) log(u 2 ) l ( log(u 1 ), log(u 2 )) = log(u 1 u 2 )A(t). (2.28) C(u 1, u 2 ) = exp{log(u 1 u 2 )A(t)}, where t = log(u 2 )/ log(u 1 u 2 ) according to (2.25). Note that this representation is not unique. By parameterising the unit simplex as 1 = {(t, 1 t) : t [, 1]}, one gets the argument log(u 1 )/ log(u 1 u 2 ) instead of log(u 2 )/ log(u 1 u 2 ) in (2.26). Several authors use this parametrization (Hürlimann [23], Abdous and Ghoudi [25]). This aspect should be considered when working with bivariate extreme-value copulas. Before discussing the properties of the Pickands dependence function A in more detail, we give the conditional copulas and the density for bivariate extreme-value copulas which we will need later, e.g. for contour plots of the parametric families or maximum likelihood estimation. The conditional copula of the random variable U 2 given U 1 = u 1 will be denoted by C 2 1 ( u 1 ) and is given by C 2 1 (u 2 u 1 ) := C(u 1, u 2 ) u 1 = C(u 1, u 2 ) u 1 (A(t) ta (t)), (2.29)

29 2.3 Extreme-value copulas 19 A(t) independence co monotonicity t Figure 2.5: Pickands dependence function: the admissible range is coloured in grey. The dashed upper line corresponds to the independence copula while the solid line denotes max{t, 1 t}, which corresponds to perfectly dependent variables. The dotted lines show two examples of typical dependence functions. where t = log(u 2 )/ log(u 1 u 2 ) and we assume that A exists. Equivalently, we get c(u 1, u 2 ) = C(u 1, u 2 ) u 1 u 2 C 1 2 (u 1 u 2 ) := C(u 1, u 2 ) u 2 = C(u 1, u 2 ) u 2 (A(t) + (1 t)a (t)). (2.3) Recall that the bivariate copula density c equals the first derivative of the copula with respect to both arguments (see (2.1)) and is thus given by [ ( )] A(t) 2 + (1 2t)A (t)a(t) (1 t)t A (t) 2 A (t), log(u 1 u 2 ) (2.31) where we again use t = log(u 2 )/ log(u 1 u 2 ) and assume the existence of A. For the parametric families discussed later this is always the case. A complete calculation of these expressions can be found in Appendix A.1. There it can also be seen that the derivation order does not play a role for the density as the Hessian matrix of the function t(u 1, u 2 ) := log(u 2 )/ log(u 1 u 2 ) is symmetric. The original definition of the Pickands dependence function (Pickands [1981]) was the following: A(t) = l (1 t, t), t [, 1], as in (2.24). By property (A2) in Proposition 2.21, a bivariate Pickands dependence function lies within the shaded area indicated in Figure 2.5. Further (A1) requires A to be convex. The solid and dashed lines in Figure 2.5 give examples of two possible Pickands dependence functions. 2.5 further illustrates the fact that the class of multivariate

30 2 Chapter 2 Preliminaries extreme-value copulas (and therewith extreme-value distributions) cannot be captured by a single parametric family. Instead any function A satisfying (A1) and (A2) is a possible Pickands dependence function. Consequently the class of bivariate extremevalue copulas is indexed by the set of convex functions defined on the unit interval. Still some important parametric families arise quite naturally and will be presented in Chapter 3. The boundaries of the Pickands dependence function A have special meanings: The upper line A(t) = 1 in Figure 2.5 corresponds to the case of independence: inserting A(t) = 1 into (2.26) yields C(u 1, u 2 ) = u 1 u 2 = Π 2 (u 1, u 2 ), the independence copula. The lower boundary A(t) = max{1 t, t} in Figure 2.5 corresponds to the case of perfect dependence: C(u 1, u 2 ) = max{u 1, u 2 } = M 2 (u 1, u 2 ). An interpretation of A is given as follows. It holds that A(t) = l (1 t, t) = 1 l(1 t, t) t [, 1]. (2.32) The argument of the Pickands dependence function thus corresponds to the argument (1 t, t) of the tail dependence functions. For l, this is exactly the vector w in (2.5). Recall that l measures the dependence along the axis (1 s(1 t), 1 st) in terms of the joint exceedance probability of both random variables. Informally spoken, the more points fall on the dependence axis or scatter around the upper right corner of the dependence axis, the higher the dependence. Now bearing in mind that A(t) = 1 corresponds to independence between the random variables, it follows from (2.32) that a higher dependence along an axis results in a smaller value of A. This is illustrated in Figure 2.6. The value A(1/2) is of special importance, as l(1, 1) measures the dependence along the diagonal. This interpretation also plays an important role in the analysis of asymmetry. In Chapter 3, popular parametric families of bivariate extreme-value copulas will be introduced.