High-Dimensional Extremes and Copulas

Transcription

1 High-Dimensional Extremes and Copulas Haijun Li Washington State University CIAS-CUFE, Beijing, January 3, 2014 Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

2 Outline 1 From Tukey to High-Dimensional Data Analysis 2 Univariate Extreme Value Theory: A Primer 3 Multivariate Extremes 4 Copula Representation of Multivariate Regular Variation 5 Tail Densities 6 Tail Densities of Vines 7 Concluding Remark 8 References Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

3 John Wilder Tukey, A chemist first and then a pure mathematician by training (PhD thesis On Denumerability in Topology in 1939) Statistician, information scientist, Presidential advisor... Figure : From Paul Halmos I Have a Photographic Memory Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

4 Tukey s Achievements Teichmüller-Tukey Lemma, equivalent to the Axiom of Choice in axiomatic set theory Tukey theory of analytic ideals, Tukey reducibility,... (A competitor to Bourbaki s approach to topology) Computation: Fast Fourier Transform Statistics: Box plot, Quenouille-Tukey Jackknife... High-dimensional data analysis: Projection Pursuit (seeking non-gaussianity) Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

5 Tukey s Collaborators and Awards Focused on obtaining results rather than collecting credits. More than 100 collaborators and 55 PhD students. Worked well with Samuel Wilks, Walter Shewhart, Claude Shannon, John von Neumann, Richard Feynman, etc. National Medal of Science, National Academy of Science, Foreign Member of the Royal Society, IEEE Medal of Honor, Samuel S. Wilks Memorial Award... Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

6 Tukey s Belief Much statistical methodology placed too great an emphasis on confirmatory data analysis rather than exploratory data analysis. (Also see Leo Breiman, Statistical Modeling: The Two Cultures, Statistical Science, 2001) People should often start with their data and then look for a theorem, rather than vice versa. Future of Data Analysis (Tukey, 1962) Data Analysis VS Mathematical Statistics?... All in all, I have come to feel that my central interest is in data analysis... Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

7 New York Times Obituary (July 28, 2000): John Tukey, 85, Statistician; Coined the Word Software A respected mathematician turned his back on mathematical proof, focusing on analyzing data instead. He legitimized that, because he wasn t doing it because he wasn t good at math, Mr. Wainer (a Tukey s former student) said. He was doing it because it was the right thing to do. (David Donoho, High-Dimensional Data Analysis: The Blessings and Curses of Dimensionality, Mathematical Challenges of the 21st Century, 2010) Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

8 Univariate Extremes X 1,..., X n are iid with df F. M n := max{x 1,..., X n } =: n i=1 X i. n i=1 X i := min{x 1,..., X n } = max{ X 1,..., X n }, and so we focus on large extremes only. Definition If there exist a non-degenerate df H, two normalizing sequences {c n } and {d n } with d n > 0, such that ( ) lim P Mn c n x = lim F n (c n + d n x) = H(x), x R, n n d n for all continuity points x of H, then F is said to be in the maximum domain of attraction of H, and this is denoted as F MDA(H) or X n MDA(H). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

9 Generalized Extreme Value Distribution (EV) Fisher-Tippett-Gnedenko Theorem If F MDA(H), then H(x; γ) = exp{ (1 + γx) 1/γ + }, x R, where γ R is called the extreme value index. After some location-scale transforms, we have Fréchet distribution: H + (x; θ) = exp{ x θ }, x > 0, θ > 0. Gumbel or double-exponential distribution: H 0 (x) = exp{ e x }, x R. (Reverse) Weibull distribution: H (x; θ) = exp{ ( x) θ }, x < 0, θ > 0. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

10 Precise Tail Variations x F := sup{x R : F(x) < 1}, the upper boundary of the support. F (u) := inf{x R : F(x) u}, 0 < u < 1, the left-continuous inverse. Gnedenko-de Haan Theorem: Fréchet MDA Case F MDA(H( ; γ)) with γ > 0 if and only if x F = and 1 F(tx) lim t 1 F(t) = x 1/γ, x > 0, where c n = 0 and d n = F (1 n 1 ). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

11 Precise Tail Variations (cont d) Gnedenko-de Haan Theorem: Gumbel and Weibull MDA Cases F MDA(H 0 ( )) if and only if x F t 0 (1 F(x))dx < for some t 0 < x F and 1 F(t + xr(t)) lim = exp{ x}, x R t x F 1 F(t) where R(t) := x F t (1 F(x))dx/(1 F(t)), t < x F and c n = F (1 n 1 ) and d n = R(c n ). F MDA(H( ; γ)) with γ < 0 if and only if x F < and 1 F(x F (tx) 1 ) lim t 1 F(x F t 1 = x 1/γ, x > 0, ) where c n = x F and d n = x F F (1 n 1 ). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

12 Example: Pareto Distribution Consider the Pareto distribution F(x) = 1 x 1/γ, x > 1, γ > 0. Since 1 F(tx) lim t 1 F(t) = x 1/γ, F MDA(H( ; γ)) with Fréchet distribution H( ; γ), γ > 0. Set normalizing constants c n = 0 and d n = F 1 (1 n 1 ) = n γ, and clearly, for x > 0, lim F n (c n + d n x) = lim (1 x 1/γ /n) n = exp{ x 1/γ }. n n Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

13 Rewrite MDA in Terms of Tail Property Without loss of generality, assume that X > 0. X F MDA(H) lim n F n (c n + d n x) = H(x). Since log x 1 x as x 1, it can be rephrased as follows lim n(1 F(c n + d n x)) = log H(x). n That is, X F MDA(H) the tail dispersion stability: ( ) X lim n P cn > x = log H(x), x > 0. n d n Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

14 A Dimension-Free Tail Property of MDA Define a (Radon) exponent measure µ(x, ] := log H(x), x R +. Rewrite: good sets B R + \{0}, µ( B) = 0, ( ) X lim n P cn > x = log H(x) n d n ( X cn lim n P n ) B = µ(b). d n }{{} Tail Dispersion Note that the exponent measure µ( ) only has an asymptotic scaling property: µ(t B) t 1/γ µ(b), good sets B R + \{0}, t. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

15 Scaling is Important for Prediction! We want: µ(t B) = t 1/γ µ(b), good sets B R + \{0}, t > 0. The tail stability and scaling allow us to estimate a tail region containing fewer observations by a tail region containing more observations, whose distribution has the same shape. Extreme Value Theory = Mechanism for relating the probabilistic structure within the range of the observed data to regions of greater extremity (Anderson, 1994). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

16 Standardization of Regular Variation (RV) Definition A random variable Z is regularly varying in the standard form if ( ) Z lim t P B = ν(b), good sets B R + \{0}, µ( B) = 0. t t Note that ν(t B) = t 1 ν(b) for any t > 0. Consider two functions c( ), d( ) such that Observe that c(tx) c(t) d(t) ψ(x) 0, x > 0, t. Tail Dispersion Standard RV {{ }} }{ {{}} }{ c(z ) c(t) Z > x = > ψ (x) d(t) t Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

17 MDA Standard RV (Klüppelberg and Resnick, 2008) Consider two functions c( ), d( ) such that c(tx) c(t) d(t) ψ(x) 0, x > 0, t. 1 If Z is standard regularly varying, then for some EV distribution H. X := c(z ) MDA(H) 2 If X MDA(H) for an EV distribution H(x), then there exists a monotone transformation c(t) such that is standard regularly varying. Z := c (X) Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

18 Remark Regular variation is also the key ingredient in weak convergence to stable laws (Meerschaert and Scheffler, 2001). Regular variation can be extended to general metric spaces to deal with functional data (Hult and Lindskog, 2006; Bingham and Ostaszewski, 2008). Extreme value distributions for data clouds in high-dimensional or functional spaces are difficult to define. Extremes emerging from high-dimensional data clouds may be best studied by exploring stability patterns of tail probability decays near data boundaries (Ledoux and Talagrand, 1991; Balkema and Embrechts, 2007). Tail risk measures (e.g., Conditional Tail Expectation) can be derived directly from regular variation properties (see, e.g., Joe and Li, 2011). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

19 von Mises Condition Let df F have a positive derivative on [x 0, x F ) for some 0 < x 0 < x F, and h F (x) = F (x)/(1 F(x)) be the hazard rate of F in a left neighborhood of x F. 1 If there exist γ R and c > 0, such that x F = then F MDA(H( ; γ/c)). { if γ 0 γ 1 if γ < 0 lim (1 + γx)h F (x) = c, (1) x x F 2 If the derivative of F is monotone in a left neighborhood of x F for some γ 0, and if F MDA(H( ; γ/c)) for some c > 0, then F satisfies (1). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

20 Examples Consider the df that satisfies that F(x) = 1 1 log x, x > e, lim (x log x)h F (x) = 1. x Since the reciprocal hazard rate of this distribution is not asymptotically linear, there is no extreme value limit for F, with affine thresholds. Note, however, that for the case where γ = 0, the von Mises condition is not necessary. For example, it is well known that the standard normal df Φ MDA(H 0 ( )), but asymptotically lim x x/h Φ (x) = 1. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

21 Laws of Small Numbers Let {Z 1,..., Z n } be iid and standard regular varying: n P (Z 1 n B) ν(b), good sets B R + \{0}, µ( B) = 0, with scaling ν(t B) = t 1 ν(b) for any t > 0. In particular, take B = (x, ) for x > 0, we have lim n P (Z 1 > nx) = ν((x, )) =: ν. n Construct a counting process of exceedances of scaled observations before time t: N n (t) = n i=1 I { i n t, Z i >nx}, where I A denotes the indicator function of set A. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

22 Dimension-Free Poisson Approximation The count of exceedances before t: N n (t) = n i=1 I { i n t, Z i >nx}, Take the expectation, we have E(N n (t)) = t np(z 1 > nx) tν, n. The counting processes (N n (t), t 0) converge to a Poisson process with rate ν, as n. The rate ν, or exponent measure ν( ) (also called the intensity measure for a good reason) encodes all the information of extremes. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

23 Multivariate Extremes X n = (X 1,n,, X d,n ), n = 1, 2,, are iid with df F(x 1,..., x d ). M n := (M 1,n,..., M d,n ), where M i,n := n j=1 X i,j, 1 i d. Notation: For any x, y R d, the sum x + y, quotient x/y, and the vector inequalities are all operated component-wise. Definition: Multivariate Extreme Value Distribution (MEV) If there exist a df G with non-degenerate margins, two normalizing sequences of real vectors {c n } and {d n } with d n > 0, such that ( ) lim P Mn c n x = lim F n (c n + d n x) = G(x), x R d, n n d n for all continuity points x of G, then F is said to be in the maximum domain of attraction of G, and this is denoted as F MDA(G) or X n MDA(G). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

24 Standardizing Margins Multivariate Scaling Let X = (X 1,..., X d ) F. X i F i : the i-th marginal df of F. G i : the i-th marginal df of G. F i MDA(G i ), where G i is the univariate, parametric EV distribution. We assume that F i is standard regularly varying. Scaling Property Emerged from Equalizing Margins 1 G i is the standard Fréchet distribution with log G i (t x i ) = t 1 log G i (x i ), x i R +. 2 Jointly, the multivariate scaling emerges from the limiting process: log G(t x) = t 1 log G(x), x R d +. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

25 A Short Proof for Multivariate Scaling: The normalizing vectors: c n = (0,..., 0), d n = (n,..., n). 1 Let k be any positive integer. We have ( lim P Mkn n n 2 On the other hand, ( lim P Mkn n n ) x = lim (F n (nx)) k = G k (x), x R d n +. ) ( x Mkn = lim P n kn x ) ( x ) = G, x R d k k +. 3 (1)+(2) yields that G k (x) = G( 1 k x) for any positive integer k. 4 Rewrite (3) as G(kx) = G 1/k (x) for x R d +. 5 (3)+(4) yields that G k/r (x) = G( r k x) for any two positive integers k, r. 6 By taking limits, we have G t 1 (x) = G(t x) for any real t > 0. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

26 Remark on scaling G(t x) = G t 1 (x), t > 0 Since margins G i s are standard Fréchet, G 1 i (u) = ( log u) 1, 0 u 1. The copula C EV of MEV distribution G enjoys the multiplicative scaling property: C EV (u t 1,..., ut d ) = Ct EV (u 1,..., u d ), (u 1,..., u d ) [0, 1] d, t > 0. This is known as the extreme value (EV) copula. If we don t standardize margins, we can decompose G into the EV copula and univariate EV distributions. We can still establish the scaling property of the EV copula (Joe, 1997, page 173). If we are not allowed to standardize margins nor to use copulas, we can establish the operator-scaling property (Meerschaert and Scheffler, 2001). But such a scaling property is not explicit. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

27 Multivariate Scaling Pickands Representation The multivariate scaling property allows us to establish a semi-parametric representation for MEV G. Let S d 1 + = {a : a = (a 1,..., a d ) R d +, a = 1}, where is a norm defined on R d. Using the polar coordinates, G and its copula can be expressed as follows: G(x 1,..., x d ) = exp b C EV (u 1,..., u d ) = exp b S d 1 + S d i d 1 i d ( ai x i ) Q(da), (a i ln u i ) Q(da), where b > 0 and Q is a probability measure defined on S d 1 + such that b a i Q(da) = 1, 1 i d. S d 1 + Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

28 Pickands Representation Approximation Method The probability measure Q is known as the spectral or angular measure. Its estimation and asymptotic properties can be found in Resnick (2007). Any probability measure Q on S d 1 + can be approximated via discrete probability measures on S+ d 1. The discretization of Q leads to a rich parametric family of max-stable multivariate Fréchet distributions (min-stable multivariate exponential distributions, including the well-known Marshall-Olkin distribution). The discretization of Q leads to a rich parametric family of EV copulas with singularity components (e.g., Lévy-frailty copulas, Mai and Scherer, 2009). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

29 Pickands Representation Positive Association Y = (Y 1,..., Y d ) is said to positively associated if E[f (Y )g(y )] Ef (Y )Eg(Y ), f, g : R d R, non-decreasing. The positive association is a strong positive dependence and has been extended to random elements in partially ordered, complete separable metric space (Lindqvist, 1988). The positive association, equivalent to FKG inequality widely used in statistical physics (Fortuin, Kastelyn, and Ginibre, 1971), is one of several basic inequalities in analyzing concentration phenomena (Ledoux and Talagrand, 1991). Theorem (Marshall and Olkin, 1983) The MEV distribution G is positively associated. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

30 Multivariate Regular Variation (MRV) Assume that df F MDA(G). X = (X 1,..., X d ) denote a random vector with distribution F and continuous, univariate margins F 1,..., F d. Without loss of generality, margins F 1,..., F d are identical. Definition (Resnick, 1987 and 2007) The df F or X is said to be multivariate regularly varying at with intensity measure ν if there exists a scaling function b(t) and a non-zero Radon measure ν( ) such that as t, ( ) X t P b(t) B ν(b), good sets B R d +\{0}, with ν( B) = 0. X is standard regularly varying if b(t) = t. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

31 Multivariate Regular Variation (MRV) Assume that df F MDA(G). X = (X 1,..., X d ) denote a random vector with distribution F and continuous, univariate margins F 1,..., F d. Without loss of generality, margins F 1,..., F d are identical. Definition (Resnick, 1987 and 2007) The df F or X is said to be multivariate regularly varying at with intensity measure ν if there exists a scaling function b(t) and a non-zero Radon measure ν( ) such that as t, ( ) X t P b(t) B ν(b), good sets B R d +\{0}, with ν( B) = 0. X is standard regularly varying if b(t) = t. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

32 Equivalently, P(X tb) lim t P(X 1 > t) = ν(b), good sets B Rd +\{0}, satisfying that ν( B) = 0. The intensity measure ν is a Radon measure with scaling property ν(tb) = t α ν(b), good sets B, bounded away from 0, where α > 0 is known as the tail index. F j, 1 j d, is regularly varying with tail index α > 0. Example: Multivariate t distribution, multivariate Pareto distributions, certain elliptical distributions,... and many many more if we use copulas and vines. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

33 MDA Standard RV (Klüppelberg and Resnick, 2008) Consider two functions c(t) = (c 1 (t),..., c d (t)), and d(t) = (d 1 (t),..., d d (t)) such that component-wise c i (tx i ) c i (t) d i (t) ψ i (x) 0, x i > 0, t. 1 If Z = (Z 1,..., Z d ) is standard regularly varying, then for some MEV distribution G. X = (c 1 (Z 1 ),..., c d (Z d )) MDA(G) 2 If X MDA(G) for an MEV distribution G(x), then there exists a component-wise monotone transformation c(t) such that is standard regularly varying. Z = (c 1 (X 1),..., c d (X d)) Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

34 Example: Heavy-Tail Case (Z 1,..., Z d ) is non-negative. For any α > 0, c i (z) := z 1/α is increasing on R +, 1 i d. (X 1,, X d ) = (c 1 (Z 1 ),..., c d (Z )) has the df F(x 1,..., x d ). Theorem (de Haan & Resnick, 1977; Marshall & Olkin, 1983) The following three statements are equivalent: 1 (Z 1,..., Z d ) is standard regular varying. 2 (X 1,, X d ) is MRV with intensity measure ν( ). 3 F MDA(G), where G(x) = exp{ ν([0, x] c )}, x R d +, is a d-dimensional distribution with Fréchet margins G i (x i ) = exp{ x α i }, 1 i d. Note that ν(tb) = t α ν(b) for any t > 0. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

35 The Copula Approach X n = (X 1,n,, X d,n ), n = 1, 2,, are iid with df F(x 1,..., x d ) that has continuous margins F 1,..., F d. The copula of F: C(u 1,..., u d ) := F(F 1 1 (u 1),..., F 1 d (u 1)), (u 1,..., u d ) [0, 1] d. Figure : Two squares are topologically equivalent. Since limiting properties are topological, the copula method should be equivalent to the standard RV method. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

36 Exponent and Tail Dependence Functions (U 1,..., U d ) := (F 1 (X n,1 ),..., F d (X n,d )). Define the exponent function: for all (w 1,..., w d ) R d +\{0}, ( P d j=1 {U j > 1 uw j } ) a(w 1,..., w d ) := lim, u 0 u with scaling a(tw 1,..., tw d ) = t a(w 1,..., w d ) for t > 0. Define the tail dependence function: ( P d j=1 {U j > 1 uw j } ) b(w 1,..., w d ) := lim, u 0 u with scaling b(tw 1,..., tw d ) = tb(w 1,..., w d ) for t > 0. The existence of the exponent function ensures the existence of the exponent and tail dependence functions of all multivariate margins of F. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

37 Various tail dependence parameters are associated with the function b( ); for example, b(1,..., 1) is known as the tail dependence coefficient. Due to the scaling property, b( ) = 0 iff b(1,..., 1) = 0, and in this case, the copula C is said to be (upper) tail independent. The notion of the exponent function a( ) can be traced back to Gumbel (1960) and Pickands (1981). It describes the full extremal dependence structure of C, including the dependence of any multivariate marginal tails. The tail dependence function b( ) is studied in Jaworski (2004, 2006), Klüppelberg, Kuhn & Peng (2008), Nikoloulopoulos, Joe & Li (2009). Note that b(w 1,..., w d ) does not necessarily cover the extremal dependence structure of a multivariate margin. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

38 Example: Archimedean Copulas Let C(u; φ) = φ( d i=1 φ 1 (u i )) be an Archimedean copula where the generator φ 1 is regularly varying at 1 with tail index β > 1. ( d ) 1/β, a(w 1,..., w d ) = w β j (w1,..., w d ) R d +\{0}. j=1 (Genest & Rivest, 1989; Barbe, Fougères & Genest, 2006) Let C have the survival copula Ĉ(u; φ) = φ( d i=1 φ 1 (u i )) with strict generator φ 1, where φ is regularly varying at with tail index θ > 0. ( d ) θ, b(w 1,..., w d ) = w 1/θ j (w1,..., w d ) R d +\{0}. j=1 (Charpentier & Segers, 2009) Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

39 Tail Dependence Function = Intensity Measure Theorem (Li and Sun, 2009) Let X = (X 1,..., X d ) be a random vector with df F and copula C. 1 If F is MRV with intensity measure ν satisfying the scaling property that ν(tb) = t α ν(b), then for w = (w 1,..., w d ), ( ( d a(w) = ν [0, w 1/α ] ) ) ( c d, b(w) = ν i=1 i i=1 ) i, ]. (w 1/α 2 If the exponent function exists and marginal dfs F 1,..., F d are (univariate) regularly varying, then F(x 1,..., x d ) = C(F 1 (x 1 ),..., F d (x d )) is MRV. The Radon measure generated by the exponent function a( ) is a rescaled version of the intensity measure ν( ). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

40 Copula Representation for MRV Ingredients for High-Dimensional MRV 1 Univariate dfs F 1,..., F d are regularly varying with tail index α 1,..., α d respectively. 2 C is a copula with (upper) tail dependence limits. Then F(x 1,..., x d ) := C(F 1 (x 1 ),..., F d (x d )) is MRV. Remark: Tail dependence properties for vine copulas are obtained using recursive constructions according to underlying graph structures (Joe, Li and Nikoloulopoulos, 2010). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

41 Copula Version of the Pickands Representation Let C be a copula with (upper) exponent function a( ). The (upper) extreme-value copula of C: C EV (u 1,..., u d ) = exp{ a( log u 1,..., log u d )}, where a(w 1,..., w d ) = S d 1 + b(w 1,..., w d ) = max 1 i d {a i w i }U(da), or S d 1 1+ min {a iw i }U(da). 1 i d The spectral measure U( ) defined on S d 1 1 is a finite measure. In contrast to the Pickands representation for intensity measures, the spectral measure U( ) does not depend on margins. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

42 Scaling Property, Again Consider the d-dimensional copula C of a random vector (U 1,..., U d ) with standard uniform margins. Recall that a(tw) = ta(w), b(tw) = tb(w), w R d +, t > 0. The well-known Euler s homogeneous theorem implies that a(w) = d j=1 a w j w j, b(w) = d j=1 b w j w j, w = (w 1,..., w d ) R d +, The partial derivatives can be interpreted as conditional limiting distributions of the underlying copula C. For example, b w j = lim u 0 P(U i > 1 uw i, i j U j = 1 uw j ). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

43 Deriving t-copula via Euler s Representation Let X = (X 1,..., X d ) have the t distribution T d,ν,σ with ν degrees of freedom and dispersion matrix Σ. The margins F i = T ν for all 1 i d, where T ν is the t distribution function with ν degrees of freedom, and that Σ = (ρ ij ) satisfies ρ ii = 1 for all 1 i d. The t-copula is defined as C t (u 1,..., u d ) = P(T ν (X 1 ) u 1,..., T ν (X d ) u d ). Derive the extreme value copula C t-ev. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

44 Partial Correlation Matrix For any i j, k j, let ρ i,k;j = ρ ik ρ ij ρ kj 1 ρ 2 ij 1 ρ 2 kj denote the partial correlations, and 1... ρ 1,j 1;j ρ 1,j+1;j... ρ 1,d;j R j = ρ 1,j 1;j... 1 ρ j 1,j+1;j... ρ j 1,d;j ρ 1,j+1;j... ρ j 1,j+1;j 1... ρ j+1,d;j ρ 1,d;j... ρ j 1,d;j ρ j+1,d;j... 1 Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

45 t-ev Copula 1 The tail dependence function of C is given by [ d ( ) b(w) = w j T d 1,ν+1,Rj ν + 1 1/ν wi + ρ ij], i j, j=1 1 ρ 2 w j ij for all w = (w 1,..., w d ) R d +. 2 The t-ev copula is given by C t-ev (u 1,..., u d ) = exp{ a(w 1,..., w d )}, w j = log u j, j = 1,..., d, with exponent [ d (wi ) a(w) = w j T d 1,ν+1,Rj ν + 1 1/ν ρ ij], i j. j=1 1 ρ 2 w j ij Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

46 Two Limiting Distributions Under some scaling conditions, C t-ev ( ) converges weakly to the Hüsler-Reiss copula as ν. As ν 0, C t-ev ( ) converges weakly to a Marshall-Olkin distribution with some linear constraints. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

47 The Tail Density Approach The notion of tail density is local and geometric (Balkema and Embrechts, 2007). Asymptotic analysis of tail risk measures often involves directly integral functionals of tail densities (Joe and Li, 2011). Geometric Risk Analysis via Tail Densities A data cloud is a realization of a random set (e.g., random vectors, spatial point processes, high-dimensional Brownian motion,...). Central limit theorems are concerned with clustering behaviors at the center of a data cloud. Extreme value theorems are concerned with dispersive stability patterns at the boundaries of a data cloud. Tail risk lives along the boundaries; it could be on the northeast boundary (max domain of attraction) or could be on the southwest boundary (min domain of attraction), or could be in other parts of boundaries. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

48 MRV Tail Densities X = (X 1,..., X d ) F that has identical continuous margins F 1,..., F d. F 1 (t) := 1 F 1 (t) denotes the survival function of F 1. Theorem (de Haan and Resnick, 1987) f (tx) Assume the density f of F exists. If λ(x) > 0 on t d F 1 (t) Rd +\{0} and uniformly on {x > 0 x = 1}, as t, then 1 F(tx) lim = λ(y)dy =: µ([0, x] c ), x R d +\{0}, t F 1 (t) [0,x] c that is, F is MRV with intensity measure µ( ). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

49 Perspective Shot of MTCJ Copula Density Consider a bivariate MTCJ copula C(x, y) = (x θ + y θ 1) 1/θ with density c(x, y) = (1 + θ)(xy) θ 1 (x θ + y θ 1) 2 1/θ, θ > 0. Approaching the origin along the ray passing though (w 1, w 2 ), w 1 > 0 and w 2 > 0, we have, as u 0, c(uw 1, uw 2 ) u 1 [(1 + θ)(w 1 w 2 ) θ 1 (w θ 1 + w θ 2 }{{ ) 2 1/θ ] } ? zmax z zmin xmin x Haijun Li High-Dimensional Extremes and xmax Copulas CIAS-CUFE, Beijing, January 3, / 72

50 Idea: Let (U 1,..., U d ) d C. When u is sufficiently small, tail density P(1 uw i U i 1 u(w i dw i ), 1 i d)/u κ dw 1 dw d u d κ P(1 uw i U i 1 uw i + d(uw i ), 1 i d) d(uw 1 ) d(uw d ) In what follows, κ = uw1 1 - uw1 + udw uw2 + udw2 1 - uw2 0 Haijun Li High-Dimensional 0 Extremes and Copulas CIAS-CUFE, 1 Beijing, January 3, / 72

51 Tail Densities of a Copula C Assume that all the necessary regularity conditions hold. Definition The upper and lower tail densities of C are defined as λ U (w) := D w C(1 uw 1,..., 1 uw d ) lim, u 0 u λ L (w) := D w C(uw 1,..., uw d ) lim, w = (w 1,..., w d ) R d + u 0 u where D w denotes the d-order partial differentiation operator and C is the survival function of C. C is upper (lower) tail independent if and only if λ U = 0 (λ L = 0). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

52 Properties Let c denote the density of C, then λ U (w) = lim u 0 u d 1 c(1 uw i, 1 i d), λ L (w) = lim u 0 u d 1 c(uw i, 1 i d), w = (w 1,..., w d ) R d + For a d-dimensional copula C, the tail densities are homogeneous of order 1 d. That is, λ U (tw) = t 1 d λ U (w) for any w R d + and t > 0. The Euler s homogeneous theorem implies that the tail densities are directionally decreasing and convex, and go down to zero at infinity. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

53 Relation with Tail Dependence Functions Recall that exponent and tail dependence functions: P(U i > 1 uw i, for some i) a(w) := lim, w = (w 1,..., w d ) R d u 0 u +. P(U i > 1 uw i, for all i) b(w) := lim, w = (w 1,..., w d ) R d u 0 u +. Theorem Recall that D w is the d-order partial differentiation operator. λ U (w) = D w b(w) = ( 1) d 1 D w a(w) for all w R d +. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

54 Relation with Tail Density of MRV Recall that λ( ) denotes the tail density of an MRV df F such that 1 F(tx) lim = λ(y)dy, x R d +. t F 1 (t) [0,x] c Once again, λ U denotes the upper tail density of the copula C of F. Theorem (Li and Wu, 2013) Let α denote the tail index of F, then λ(w 1,..., w d ) = α d (w 1 w d ) α 1 λ U (w α 1,..., w α d ) = J(w α 1,..., w α d ) λ U (w α 1,..., w α d ), where J(w α 1,..., w α d ) is the Jacobian determinant of the homeomorphism y i = w α i, 1 i d. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

55 Archimedean Tail Densities Let C(u; φ) = φ( d i=1 φ 1 (u i )) be an Archimedean copula where the Laplace transform φ. Lower Tail Density If φ is regularly varying at with tail index θ > 0, then d ( λ L (w) = 1 + i 1 )( d ) 1 1/θ ( d ) θ d. w i w 1/θ θ i i=1 Upper Tail Density i=1 i=1 If φ 1 is regularly varying at 1 with tail index β > 1, then λ U (w) = d ( d ) β 1 ( d ) d 1/β. ((i 1)β 1) w i w β i i=1 i=1 i=1 Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

56 Lower Archimedean Tail Density Figure 2: graphs for lower tail case Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

57 Upper Archimedean Tail Density Figure 1: graphs for upper tail case Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

58 t Tail Density Consider a d-dimensional symmetric t distribution with cdf f (x; ν, Σ) = ν+d Γ( 2 ) [ Γ( ν 2 )(νπ)d/2 Σ ν (x T Σ 1 x) ] ν+d 2, x R d where ν > 0 is the degree of freedom, and Σ is a d d dispersion matrix. Tail Density of t-copula λ U (w) = λ L (w) = π ν+d d 1 2 Σ 1 2 ν 1 d Γ( 2 ) for any w R d +. } {{ Γ( ν+1 2 ) } ζ [(w 1 ν ) T Σ 1 w 1 ν ] ν+d 2 d i=1 w ν+1 ν i. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

59 Rewrite It as a Norm-Based Tail Density Define a norm w := (w T Σ 1 w) 1/2, w R d +. Let J(w 1/ν 1,..., w 1/ν d ) be the Jacobian determinant of topologically invariant transform y i = w 1/ν i, 1 i d. Rewrite the upper tail density of a t copula: λ U (w 1,..., w d ) = ζ d i=1 For the symmetric t distribution itself, (de Haan and Resnick, 1987). w ν+1 ν i ((w 1/ν ) T Σ 1 w 1/ν ) ν+d 2 = ζν d J(w 1/ν 1,..., w 1/ν d ) w 1/ν ν d λ(w 1,..., w d ) = ζν d w ν d Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

60 Vine Copula C of (U 1,..., U d ) For any u = (u 1,..., u d ) [0, 1] d, define u S = (u i, i S), S {1,..., d}. Ingredients for Vines {c i,j, 1 i < j d} = A set of the densities of bivariate linking copulas. For any S {1,..., d}, define the S-marginal density c S := c S (u S ). Define the conditional distribution of U k given U S = u S C k S := C k S (u k u S ), k / S. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

61 D-Vine C The density c {1,...,d} of C is constructed recursively as follows. D-Vine Construction (Bedford and Cooke, 2001 and 2002) 1 Baseline: For any 1 i d 1, the density of the {i, i + 1} margin is c i,i+1. 2 Recursion: c {1,...,d} ( ) c {1,...,d 1} c {2,...,d} = c 1,d C1 2,...,d 1, C c d 2,...,d 1. {2,...,d 1} c {2,...,d 1} c {2,...,d 1} Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

62 The Recursion of D-Vine Lower Tail Densities For any S {1,..., d}, λ L S (w S) = lower tail density of the S-margin C S. For any S {1,..., d} and k / S, define the lower tail conditional distribution of U k given U S = u S : t k S (w k w S ) := lim u 0 C k S (uw k uw S ). Theorem Fix S = {2,..., d 1}. If all the bivariate baseline linking copulas have lower tail dependence, then λ L (w) λ L S (w S) = c 1,d ( t1 S (w 1 w S ), t d S (w d w S ) ) λ L {1} S (w) λ L S (w S) λ L {d} S (w) λ L S (w S). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

63 Examples The 3-dimensional D-vine: λ L (w 1, w 2, w 3 ) = λ L 12 (w 1, w 2 ) λ L 23 (w 2, w 3 ) c 13 (t 1 2 (w 1 w 2 ), t 3 2 (w 3 w 2 )). The 4-dimensional D-vine: λ L (w 1,w 2, w 3, w 4 ) = λ L 12 (w 1, w 2 ) λ L 23 (w 2, w 3 ) λ L 34 (w 3, w 4 ) c 13 (t 1 2 (w 1 w 2 ), t 3 2 (w 3 w 2 )) c 24 (t 2 3 (w 2 w 3 ), t 4 3 (w 4 w 3 )) c 14 (t 1 23 (w 1 w 2, w 3 ), t 4 23 (w 4 w 2, w 3 )). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

64 Cautionary Remark What happens if some or all bivariate baseline linking copulas of a D-vine C are tail independent? Joe et al (2010) has a partial answer: C must be tail independent, e.g., C(uw i, 1 i d) u κ h(w), as u 0, κ > 1. Some margins of C can still be tail dependent. But can we quantify the order of scaling, e.g., κ, in the case of tail independence of a vine copula? Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

65 Seeking Recursions of RV Near Data Boundaries Consider the 3-dimensional D-vine c(uw 1,uw 2, uw 3 ) = c 12 (uw 1, uw 2 ) c 23 (uw 2, uw 3 ) c 13 (C 1 2 (uw 1 uw 2 ), C 3 2 (uw 3 uw 2 )), when u is small. As functions of u, the regular variations of constructs c 12, c 23, C 1 2, C 3 2, and c 13 should yield the regular varying property of c. Example: If the baseline linking copulas C 12 and C 23 are Morgenstern copulas, and the linking copula C 13 is a t copula, then c 12 (uw 1, uw 2 ) and c 23 (uw 2, uw 3 ) are asymptotically constant as u 0 and C 1 2 (uw 1 uw 2 ) uh 1 2 (w 1, w 2 ), C 3 2 (uw 3 uw 2 ) uh 3 2 (w 2, w 3 ), as u 0. Thus, as u 0, That is, κ = 2. c(uw 1, uw 2, uw 3 ) u 1 h(w 1, w 2, w 3 ). Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

66 Concluding Remark: Predictability vs. Interpretability In analyzing extremes, the goal is to develop accurate prediction rather than to provide good interpretability. In analyzing high-dimensional data clouds, margins are of multiple scales and it is entirely possible that only some multivariate margins converge under possibly different surface/hyperplane thresholdings (Balkema and Embrechts, 2007). Occam s Razor is often of little use in high-dimensional dependence analysis. Simple and interpretable models do not make the most accurate predictions. Sophisticated and flexible models are called for. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

67 If all a man has is a hammer,... Leo Breiman, Statistical Modeling: The Two Cultures, Statistical Science (2001) An old saying: If all a man has is a hammer, then every problem looks like a nail. But the trouble for statisticians lately is that some of the new problems have stopped looking like nails. Complex big data pose significant challenges. Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

68 References Balkema, G. and Embrechts, P.: High Risk Scenarios and Extremes: A geometric approach. European Mathematical Society, Zürich, Switzerland, Barbe, P., Fougères, A. L., Genest, C.: On the tail behavior of sums of dependent risks. ASTIN Bulletin, 2006, 36(2): Bedford, T. and Cooke, R. M.: Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial Intelligence, 2001, 32: Bedford, T. and Cooke, R. M.: Vines - a new graphical model for dependent random variables. The Annals of Statistics, 2002, 30: Charpentier, A. and Segers, J.: Tails of multivariate Archimedean copulas. Journal of Multivariate Analysis, 2009, 100: Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

69 References de Haan, L. and Resnick, S.I.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheor. Verw. Gebiete, 1977, 40: de Haan, L. and Resnick, S.I.: On regular variation of probability densities. Stochastic Process. Appl., 1987, 25: Genest, C. and Rivest, L.-P.: A characterization of Gumbel s family of extreme value distributions. Statistics and Probability Letters, 1989, 8: Hua, L. and Joe, H.: Tail order and intermediate tail dependence of multivariate copulas. Journal of Multivariate Analysis, 2011, 102: Hua, L., Joe, H. and Li, H.: Relations between hidden regular variation and tail order of copulas (under review). Technical report, Department of Mathematics, Washington State University, Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

70 References Hüsler, J. and Reiss, R.-D.: Maxima of normal random vectors: between independence and complete dependence. Statistics & Probability Letters, 1989, 7(4): Jaworski, P.: On uniform tail expansions of bivariate copulas. Applicationes Mathematicae, 2004, 31(4): Jaworski, P.: On uniform tail expansions of multivariate copulas and wide convergence of measures. Applicationes Mathematicae, 2006, 33(2): Jaworski, P.: Tail Behaviour of Copulas, Chapter 8 in Copula Theory and Its Applications, Eds. P. Jaworski, F. Durante, W. Härdle, and T. Rychlik, Lecture Notes in Statistics 198, Springer, New York, Joe, H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London, Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

71 References Joe, H. and Li, H.: Tail risk of multivariate regular variation. Methodology and Computing in Applied Probability, 2011, 13: Joe, H., Li, H. and Nikoloulopoulos, A.K.: Tail dependence functions and vine copulas. Journal of Multivariate Analysis, 2010, 101: Klüppelberg, C., Kuhn, G. and Peng, L.: Semi-parametric models for the multivariate tail dependence function the asymptotically dependent. Scandinavian Journal of Statistics, 2008, 35(4): Klüppelberg, C. and Resnick, S.: The Pareto copula, aggregation of risks, and the emperor s socks. J. Appl. Probab., 2008, 45(1): Kurowicka, D. and Cooke, R.: Uncertainty Analysis with High Dimensional Dependence Modelling. Wiley, New York, Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

72 References Ledoux, M., and Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Li, H. and Sun, Y.: Tail dependence for heavy-tailed scale mixtures of multivariate distributions. J. Appl. Prob., 2009, 46 (4): Li, H. and Wu, P.: Extremal dependence of copulas: A tail density approach. Journal of Multivariate Analysis, 2012, 114: Marshall, A. W., and Olkin, I.: Domains of attraction of multivariate extreme value distributions. Annuals of Probability, 1983, 11: Meerschaert, M. M. and Scheffler, H.-P.: Limit Distributions for Sums of Independent Random Vectors. John Wiley & Sons, Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72

73 References Mai, J.-F. and Scherer, M.: Reparameterizing Marshall-Olkin copulas with applications to sampling. Journal of Statistical Computation & Simulation, 2009, Nikoloulopoulos, A. K., Joe, H. and Li, H.: Extreme value properties of multivariate t copulas. Extremes, 2009, 12: Resnick, S.: Extreme Values, Regularly Variation, and Point Processes, Springer, New York, Resnick, S.: Hidden regular variation, second order regular variation and asymptotic independence. Extremes, 2002, 5: Resnick, S.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York, Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January 3, / 72