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1 Department of Mathematics Technical Report Extreme Value Properties of Multivariate t Copulas Aristidis K Nikoloulopoulos, Harry Joe, Haijun Li April 2008 Postal address: Department of Mathematics, Washington State University, Pullman, WA Voice: Fax: info@mathwsuedu URL:

2 Extreme Value Properties of Multivariate t Copulas Aristidis K Nikoloulopoulos Harry Joe Haijun Li Abstract The extremal dependence behavior of t copulas is examined and their extreme value limiting copulas, called the t-ev copulas, are derived explicitly using the tail dependence functions As two special cases, the Hüsler-Reiss and the Marshall-Olkin distributions emerge as limits of the t-ev copula as the degrees of freedom go to infinity and zero respectively The t copula and its extremal variants attain a wide range in the set of bivariate tail dependence parameters Mathematics Subject Classification: 62H20, 91B30 Key Words and Phrases: Tail dependence function, extreme value, t copula 1 Introduction It has been well documented see, for example, Demarta and McNeil, 2005; McNeil et al, 2005) that the t copulas are generally superior to the Gaussian copulas in the context of modeling multivariate financial return data, due in great part to the ability of t copulas to capture the tail dependence among extreme values This paper is motivated from studying the range of the set of bivariate tail dependence parameters for t copulas and showing that that sharp bounds for a triplet of bivariate tail dependence parameters are almost always reached Let F denote the distribution function of a random vector X = X 1,, X d ) with continuous margins F 1,, F d The copula C of F can be uniquely expressed as Cu 1,, u d ) = F F 1 1 u 1 ),, F 1 d u d )), u 1,, u n ) [0, 1] d, 11) where F 1 i, 1 i d, are the quantile functions of the margins That is, the copula C can be thought of as the distribution function of the marginally transformed random vector U 1,, U d ) = F 1 X 1 ),, F d X d )) with standard uniform marginal distributions see, eg, Joe, 1997) For example, a Gaussian copula is obtained via 11) from a multivariate standard normal distribution with correlation matrix R, and a t copula is obtained via 11) from a multivariate t distribution with ν degrees of freedom df) and dispersion matrix Σ The Gaussian and t copulas inherit the dependence structure of multivariate normal and t distributions Supported by NSERC Discovery Grant {akn,harry}@statubcca, Department of Statistics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 lih@mathwsuedu, Department of Mathematics, Washington State University, Pullman, WA 99164, USA 1

3 One important dependence property is the tail dependence, which describes the dependence in the lower or upper tails of a distribution In the bivariate case, the lower and upper tail dependence parameters are defined as follows see Chapter 2 of Joe, 1997), λ L Pr{X i Fi 1 u), X j Fj 1 u)} ij = lim, λ U Pr{X i > Fi 1 u), X j > Fj 1 u)} ij = lim, u 0 u u 1 1 u for any 1 i < j d For reflection) symmetric distributions such as the t distributions, λ ij := λ L ij = λu ij If, furthermore, the univariate margins are identical, then we have C ij u, u) F ij t, t) λ ij = lim = lim u 0 u t F i t), where C ij and F ij denote the i, j) bivariate margin of C and F respectively It is well-known that the Gaussian copula has no tail dependence ie, λ ij = 0 for any i j), whereas the t copula enjoys the full spectrum of tail dependence Embrechts et al, 2002), ) ν ρij λ ij = 2T ν+1, 12) 1 + ρij where T m denotes the univariate t distribution function with m df, and ρ ij is the correlation coefficient of the i, j) margin That is, λ ij is increasing in ρ ij, from 0 to 1 as ρ ij goes from 1 to 1 Let {λ ij : 1 i j d} be the set of bivariate lower or upper) tail dependence parameters of a d-dimensional copula C The study of the range of triplets λ ij, λ jk, λ ik ), for i, j, k distinct, is equivalent to the study of the range of λ ik given λ ij, λ jk A copula family with a wider range would have more versatility in modeling tail behavior For t copulas, the inequalities for a triplet of tail dependence parameters are related to inequalities for a triple of correlation coefficients ρ ij, ρ jk, ρ ik ) We will refer to the parameters of dispersion matrix of the multivariate t distribution as correlations even though second moments exists only for ν > 2 Based on partial correlations see, eg, Kurowicka and Cooke, 2006), for elliptical copulas that include t copulas, the correlation coefficients ρ ik, 1 i < k d, satisfy 1 ρ ik;j 1 or ρ ij ρ jk 1 ρ 2 ij )1 ρ2 jk ) ρ ik ρ ij ρ jk + 1 ρ 2 ij )1 ρ2 jk ), i < j < k 13) Inequalities for the bivariate tail dependence parameters have also been established Theorem 314, Joe, 1997), max{0, λ ij + λ jk 1} λ ik 1 λ ij λ jk, i < j < k 14) Unlike 13), however, the sharpness of 14) had not been shown In this paper, we establish the sharpness of the inequality 14) using the t copulas and min-stable trivariate exponential distributions The sharpness of the bounds in 14) can be achieved in most cases through limiting t copulas, and this motivates us to study the extreme value t copulas t-ev copulas), which preserve the tail dependence parameters of the t copula 2

4 see Theorem 68 of Joe, 1997) It should be mentioned that Demarta and McNeil 2005) derived the EV copula limit for the bivariate t copulas using the standard techniques of Extreme Value Theory To ease the cumbersome notational burden in higher dimensions, we introduce the tail dependence and conditional tail dependence functions to derive the t-ev copulas This tool box of tail dependence functions, which is of interest in its own right, is particularly effective for the tail dependence analysis of multivariate t copulas As two limiting cases, the t-ev copulas yield the Hüsler-Reiss distribution and the Marshall-Olkin distribution as the df ν goes to infinity and zero respectively The remainder of the paper proceeds as follows Section 2 introduces the multivariate tail dependence functions, and derives the t-ev copula and its limiting cases Section 3 establishes the sharpness of the inequality 14) Also in Section 3, several distributions are compared in the ranges of triplets of tail dependence parameters Section 4 concludes with some discussion and further research 2 Extreme Value Limits of Multivariate t Copulas Consider the d-dimensional copula C of a random vector U 1,, U d ) with the standard uniform margins and continuous second-order partial derivatives Let I := {1,, d} denote the index set For any S I, let C S denote the copula of the S -dimensional margin U i, i S) Define the lower tail dependence functions b S ) of C, S I, as follows, C S uw i, i S) b S w i, i S) := lim, w = w 1,, w d ) R d u 0 u + 21) Note that b S w) = w if S = 1 for univariate margins, and b ij 1, 1) = λ L ij becomes the lower tail dependence parameter of the i, j) margin Analogous upper tail dependence functions could be defined, but they are the same for t copulas because of their reflection symmetry The following elementary properties of the function b S ), S I, can be easily verified 1 b S w i, i S) = 0 if least one w i is zero, i S 2 b S w i, i S) is non-decreasing or increasing) in w i S 3 For any fixed t > 0, b S tw i, i S) = lim u 0 C S utw i, i S)/u = t lim u 0 C S utw i, i S)/ut) = tb S w i, i S), and thus b S ) is homogeneous of order 1 Since bw) := b I w) is differentiable and homogeneous of order 1, the well-known Euler s homogeneous theorem implies that see, eg, Wilson, 1912) bw) = d b w j w j, w = w 1,, w d ) R d +, 22) 3

5 where the partial derivatives b/ w j are homogeneous of order 0 and bounded For any 1 j d, let I j := I {j} Observe from 21) that for any 1 j d, b w j = lim u 0 Pr{U 1 uw 1,, U j 1 uw j 1, U j+1 uw j+1,, U d uw d U j = uw j } = lim u 0 Pr{U i uw i, i I j U j = uw j } = lim u 0 C Ij {j}uw i, i I j uw j ), 23) where the notation C S1 S 2 refers to the conditional distribution of {U i : i S 1 } given {U k : k S 2 } The partial derivatives b/ w j are called the conditional lower tail dependence functions of copula C To express the EV copula of C in terms of b S ), we need the following function, aw) := S I,S It follows from 21) and the inclusion-exclusion relation that 1 aw) = lim u 0 u S I,S 1) S 1 C S uw i, i S) 1 = lim u 0 u Pr{U i uw i, for some i I} = lim 1 = lim 1 Pr{1 Ui 1 uw i, i I} ), u 0 u which implies the approximation 1) S 1 b S w i, i S) 24) u u 0 1 Pr{1 U i 1 uw i, i I} 1 u aw), u 0 1 Pr{Ui > uw i, i I} ) Let C denote the copula of 1 U 1,, 1 U d ) The lower EV limit of C is the same as the upper EV limit of C, which is given by lim n C ) n u 1/n 1,, u 1/n d ) Chapter 6, Joe, 1997) For sufficiently large n, u 1/n j = exp{n 1 log u j } 1 + n 1 log u j, so that with ũ j = log u j, as n C ) n u 1/n 1,, u 1/n d ) [ 1 n 1 aũ 1,, ũ d ) ] n exp{ aũ1,, ũ d )}, Theorem 21 Let C EV denote the lower EV copula of a copula C, then C EV u 1,, u d ) = exp{ a log u 1,, log u d )}, where a ) can be evaluated from 21)-24) Next we apply the above for the multivariate t copulas Let X = X 1,, X d ) have the t distribution with ν df and dispersion matrix Σ Since the increasing location-scale marginal transforms convert X to the t distribution with identical margins, we assume, for expositional simplicity, that the margins F i = T ν for all 1 i d, where T ν is the t distribution function 4

6 with ν df, and that Σ = ρ ij ) satisfies ρ ii = 1 for all 1 i d The conditional tail dependence function in 23) can be rephrased as b = lim Pr{X i Tν 1 uw i ), i I j X j = Tν 1 uw j )}, 1 j d w j u 0 Consider z = Tν 1 uw) Since the lower tail of the density T νz) behaves like α ν ν z ν+1) where α ν = ν ν 1)/2 Γ ν + 1)/2 ) Γν/2) πν, the distribution T ν z) behaves like α ν z ν as z Hence T 1 ν u) α ν /u) 1/ν and z uw/α ν ) 1/ν as u 0 Thus, the conditional tail dependence function of 23) for the t copula is expressed as b w j = lim u 0 Pr{X i uw i /α ν ) 1/ν, i I j X j = uw j /α ν ) 1/ν }, 1 j d 25) To obtain the conditional distribution in 25) from the multivariate t distribution, we need the following result see, eg, Kotz and Nadarajah, 2004) Let T d,ν,σ denote the distribution function of a d-dimensional t distribution with df ν and dispersion matrix Σ Lemma 22 Let X = X 1, X 2 ) have the multivariate t distribution T d,ν,σ with ) Σ 11 Σ 12 Σ =, Σ 22;1 = Σ 22 Σ 21 Σ 1 11 Σ 21 Σ Σ 12, 22 where X 1 and X 2 are k and d k dimensional respectively Then ) ν + k Pr{X 2 x 2 X 1 = x 1 } = T d k,ν+k,σ22;1 ν + x 1 Σ 1 11 x x 2;1 1 where x 2;1 = x 2 Σ 21 Σ 1 11 x 1 Combining 25) and Lemma 22, we have b ν ) = T d 1,ν+1,Σj + 1 w 1 /w j ) 1/ν + ρ 1j,, ) ν + 1 w j 1 /w j ) 1/ν + ρ j 1,j, w j ) ν + 1 w j+1 /w j ) 1/ν + ρ j+1,j,, )) ν + 1 w d /w j ) 1/ν + ρ dj ν ) ) = T d 1,ν+1,Σj + 1 w i /w j ) 1/ν + ρ ij, i I j ) = T d 1,ν+1,Rj ν + 1 1/ν wi + ρ ij), i I j, 26) 1 ρ 2 w j ij where 1 ρ 2 1j ρ 1,j 1 ρ 1jρ j 1,j ρ 1,j+1 ρ 1jρ j+1,j ρ 1d ρ 1jρ dj Σ j = ρ 1,j 1 ρ 1jρ j 1,j 1 ρ 2 j 1,j ρ j 1,j+1 ρ j 1,jρ j+1,j ρ j 1,d ρ j 1,jρ dj ρ 1,j+1 ρ 1jρ j+1,j ρ j 1,j+1 ρ j 1,jρ j+1,j 1 ρ 2 j+1,j ρ j+1,d ρ j+1,jρ dj ρ 1d ρ 1jρ dj ρ j 1,d ρ j 1,jρ dj ρ j+1,d ρ j+1,jρ dj 1 ρ 2 dj 5

7 and 1 ρ 1,j 1;j ρ 1,j+1;j ρ 1,d;j ρ 1,j 1;j 1 ρ j 1,j+1;j ρ j 1,d;j R 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 27) with ρ i,k;j = ρ ik ρ ij ρ kj 1 ρ 2 ij 1 ρ 2 kj, i j, k j, being the partial correlations Substituting 26) in 22) and 24), we obtain the tail dependence function and the EV limit of the t copula Theorem 23 Let C be the copula of a multivariate t distribution with ν df and dispersion matrix Σ = ρ ij ) with ρ ii = 1 for 1 i d 1 The tail dependence function of C is given by [ d bw) = w j T d 1,ν+1,Rj ν ρ 2 ij for all w = w 1,, w d ) R d + wi w j ) 1/ν + ρ ij], i I j, 2 The t-ev copula, obtained by the EV limit of the t copula, is given by with exponent C EV u 1,, u d ) = exp{ aw 1,, w d )}, w j = log u j, j = 1,, d, aw) = d [ ) w j T d 1,ν+1,Rj ν + 1 1/ν wi ρ ij], i I j 28) 1 ρ 2 w j ij Proof We only need to derive 28) It follows from 22), 24), and 26) that = = lim u 0 = lim u 0 aw) = S I,S S I,S 1) S 1 S I,S 1) S 1 j S d w j 1 1) S 1 b S w i, i S) b S w j w j j S { ) 1/ν uwi ) } 1/ν uwj w j Pr X i, i S j Xj = S:j S, S 2 α ν { ) 1/ν 1) S 2 uwi ) } 1/ν uwj Pr X i, i S j Xj = α ν α ν α ν 6

8 where S j = S {j} I j The inclusion-exclusion principle implies that d ) aw) = w j T d 1,ν+1,Rj ν + 1 1/ν wi + ρ ij), i I j 1 ρ 2 w j ij d ) = w j T d 1,ν+1,Rj ν + 1 1/ν wi ρ ij), i I j, 1 ρ 2 w j ij where T d 1,ν+1,Rj ) is the survival function of the distribution function T d 1,ν+1,Rj It is known that the elliptical distribution functions are increasing in ρ ij in the sense of the concordance ordering; see, eg, Theorem 221 of Joe 1997) The tail dependence function b ) of a t copula and its t-ev copula inherit this monotonicity property It is also evident from Theorem 23 that b1, 1,, 1) respectively a1, 1,, 1)) are decreasing respectively increasing) in df ν Note, however, that as our numerical results show, for any fixed w and ρ ij, the monotonicity of aw) with respect to ν does not hold in general Note that the t-ev copula family given in Theorem 23 have positive dependence only even if it is parametrized by a correlation matrix The independence copula obtains only in the limit as ν Next we discuss the limits of 28) as ν 0 and ν Let X = X 1,, X d ) have the multivariate standard normal distribution with correlation matrix R = ρ ij ) Hüsler and Reiss 1989) derived the EV limit of X under the limiting constraint ρ ij n) = 1 4δ 2 ij 1, as n, 29) log n for positive parameters δ ij, i j, with δ ij = δ ji, and some other constraints The Hüsler-Reiss copula C HR can be rephrased as a min-stable multivariate exponential distribution as follows see page 183 of Joe, 1997), for any non-negative w 1,, w d, The exponent function a ) is given by aw; δ ij, 1 i, j d) = C HR e w 1,, e w d ) = exp{ aw; δ ij, 1 i, j d)} 0 d w j + S I, S 2 1) S 1 r S w i, i S; δ ij ) i,j S ), 210) where for S = {i 1,, i s } I, S = s 2, wis r S w i, i S; δ ij ) i,j S ) = Φ s 1,ΓS,is δi 1 j,i s + δ i j,i s log y ), 1 j s 1 dy, 211) 2 w ij and Φ s 1,ΓS,j ) is the survival function of the multivariate normal distribution with correlation matrix Γ S,j = ϱ i,k;j ) whose i, k) entry equal to, i, k S {j}, ϱ i,k;j := δ 2 ij + δ 2 kj 2δ 1 ij δ 1 kj 7 δ 2 ik, 212)

9 with δii 1 is defined as zero for all i As our next result shows, the Hüsler-Reiss copula C HR emerges as the limiting distribution of the t-ev copula as ν with ρ ij 1 at appropriate rates Theorem 24 Let Cν EV ) be the t-ev copula obtained in Theorem 23 with dispersion matrix Σ = ρ ij ν)), where ρ ij ν) = 1 2δij 2 /ν and the parameters δ ij are the same as these in 29) Then Cν EV ) converges weakly to the Hüsler-Reiss copula C HR ) as ν Proof For any S I, let CS,ν EV of C EV ) CHR S )) denote the S -dimensional copula of the margin ν ) C HR )) with component indexes in S Let b S,ν ), S I, denote the tail dependence function of t-ev copula C EV ν ) as defined in 21) Observe that r S w i, i S; δ ij, i, j S) in 211) is homogeneous of order 1 It is clear from 24) and 210) that we only need to show that r S ; δ ij, i, j S) of C HR ) is the limit of the tail dependence function of C EV ν ); that is, for any S I, r S w i, i S; δ ij ) i,j S ) = lim ν b S,νw i, i S), w = w 1,, w d ) R d + 213) For this, let R S,j, j S, denote the partial correlation matrix of CS,ν EV ) as defined in 27), and let Γ S,j denote the correlation matrix of CS HR ) as defined in 212) It is easy to verify that for any 1 j d, ρ ik ν) ρ ij ν)ρ kj ν) 1 δ 2 ij ρ 2 ij ν) 1 ρ 2 kj ν) + δ 2 kj 2δ 1 ij δ 1 kj δ 2 ik = ϱ i,k;j as ν Thus, R S,j converges entry-wise to Γ S,j Let S = {i 1,, i s } I, S = s 2 From 26) and 211), we have [ ) b S,ν = F s 1,ν+1,RS,is ν + 1 1/ν wi + ρ iis], i S {i s } w is 1 ρ 2 w is ii s [ wij ) = F s 1,ν+1,RS,is ν + 1 1/ν ρ ijis], 1 j s 1 1 ρ 2 w is i j i s r S w i, i S; δ ij ) i,j S ) = Φ s 1,ΓS,is δi 1 w j,i s + δ i j,i s log w ) i s, 1 j s 1 is 2 w ij Observe that as ν, wis ) [ ν + 1 1/ν wis ρ ijis) 1 ρ 2 w ij w ij i j i s log w is w ij + 2δ 2 i j,i s ν 2 ν ν 4δ 2 i j,i s ) ) 1/ν 1] + 2δ 2 i j,i s ν 2 ν 4δi 2 j,i s δi 1 j,i s + δ i j,i s log w i s 2 w ij 8

10 Therefore Since b S,ν / w is b S,ν r S w i, i S; {δ ij } i,j S ) = lim w is ν w is is bounded, the bounded convergence theorem implies that and 213) holds r S w i, i S; δ ij ) i,j S ) = lim ν wis 0 b S,ν x is dx is = b S,ν w i, i S), Remark 25 Note that our method of tail dependence functions also yields a more explicit expression for the Hüsler-Reiss copula It follows from Theorems 23 and 24 that the exponent function for the Hüsler-Reiss copula can be expressed as follows, aw; δ ij, 1 i, j d) = d w j Φ d 1,ΓI,j δij 1 + δ ij 2 log w ) j, i I j w i This homogeneous representation provides better insight into understanding the structure of the Hüsler-Reiss copula: The rates of changes for its exponent function are driven by normal distributions Hüsler and Reiss 1989) obtained this expression for the bivariate case: aw 1, w 2 ) = w 1 Φ δ 1 + δ )) 2 log w1 + w 2 Φ δ 1 + δ )) w 2 2 log w2 w 1 Their method, however, seems too cumbersome to extend this to higher dimensions in the form of 22) Our form is easier to use in higher dimensions; only a function for the multivariate normal cumulative distribution function is needed, and not another integral As ν 0, the rates of changes of the exponent function 28) of the t copula depends on the arguments w i, 1 i d, only through their ranks and thus the singularities appear Let F d,σ denote the distribution function of a d-dimensional Cauchy distribution with dispersion matrix Σ Since w j /w i ) 1/ν goes to zero or infinity, as ν 0, depending on whether w j < w i or w j > w i, the exponent 28) as ν 0 becomes a 0 w) = d w j F d rwj ),R Sj w);j ρ ij 1 ρ 2 ij, i S j w), 214) where F 0 := 1, rw j ) denotes the rank of w j among w 1,, w d, S j w) = {i : w i > w j }, and R Sj w);j is the sub-matrix of 27) by retaining the columns and rows with indexes in S j w) Observe that the exponent a 0 w) is a linear function of w i, 1 i d, with the coefficient depending on its rank, and thus, as the next example shows, resembles the exponent functions of Marshall-Olkin distributions; see Marshall and Olkin 1967a,b) Example 26 Consider the trivariate case of a 0 ) Let ρ ij α ik;j ρ ij, ρ kj ) = F 2,RSj w);j 1 ρ 2 ij, ρ kj 1 ρ 2 kj, if S j w) = {i, k}, 9

11 ρ ij α i;j ρ ij ) = F 1,RSj if S w);j j w) = {i} 1 ρ 2 ij For w 1 < w 2 < w 3, a 0 w) = α 23;1 ρ 21, ρ 31 )w 1 + α 3;2 ρ 32 )w 2 + w 3, 215) In contrast, the survival function of a trivariate Marshall-Olkin distribution with non-negative parameters {β S : S {1, 2, 3}} can be written as G MO w 1, w 2, w 3 ) = Pr{W 1 > w 1, W 2 > w 2, W 3 > w 3 } = exp{ a MO w 1, w 2, w 3 )}, where the exponent function is given by a MO w 1, w 2, w 3 ) = 3 β j w j + 1 i<j 3 β ij w i w j ) + β 123 w 1 w 2 w 3 ), with β ij = β ji Here and in the sequel denotes the maximum For w 1 < w 2 < w 3, we have a MO w 1, w 2, w 3 ) = β 1 w 1 + β 2 + β 12 )w 2 + β 3 + β 13 + β 23 + β 123 )w 3 216) The two exponent functions coincide if we set β 1 = α 23;1 ρ 21, ρ 31 ), β 2 + β 12 = α 3;2 ρ 32 ), β 3 + β 13 + β 23 + β 123 = 1 Since the coefficients of both a 0 ) and a MO ) depend on the ranks of their corresponding arguments, the other equations can be obtained by simply interchanging any two indexes: β 2 = α 13;2 ρ 12, ρ 32 ), β 3 = α 12;3 ρ 13, ρ 23 ), β 2 + β 32 = α 1;2 ρ 12 ) β 1 + β 12 = α 3;1 ρ 31 ), β 1 + β 13 = α 2;1 ρ 21 ), β 3 + β 13 = α 2;3 ρ 23 ), β 3 + β 23 = α 1;3 ρ 13 ), β 1 + β 31 + β 21 + β 123 = 1, β 2 + β 12 + β 32 + β 123 = 1 This system of equations yields the non-negative solutions α 13;2 ρ 12, ρ 32 ) 0, and For example, β 12 = α 3;2 ρ 32 ) β 123 = [1 α 2;3 ρ 23 )] [α 1;3 ρ 13 ) α 12;3 ρ 13, ρ 23 )] 0, due to the -monotonicity of the distribution function F 3,Σ The other parameters can be derived by simply exchanging any two indexes, and thus 215) can be written as 216) The Marshall-Olkin distribution provides a stochastic representation for the limiting distribution of the t copula as ν 0 It follows from Marshall and Olkin 1967b) that G MO ) with exponent 216) is the survival function of the following random variables, W 1 = min{y 1, Y 12, Y 13, Y 123 }, W 2 = min{y 2, Y 12, Y 23, Y 123 }, W 3 = min{y 3, Y 23, Y 13, Y 123 }, where Y S s are independent exponential random variables with EY S ) = 1/β S for S {1, 2, 3} The singularities of the t copula as ν 0 are apparent from such a representation 10

12 Using the similar method and notations as in Example 26, the Marshall-Olkin representation for the t-ev copula as ν 0 can be extended to the d-dimensional case The survival function of a Marshall-Olkin distribution can be written as G MO w 1,, w d ) = Pr{W 1 > w 1,, W n > w d } = exp{ a MO w)} where a MO w) = β S i S w i ), 217) for some non-negative parameters β S s S: S I Theorem 27 The multivariate t-ev copula with exponent 214) converges weakly as ν 0 to a Marshall-Olkin distribution G MO with exponent of the form 217) Proof We first introduce the notation, ρ ij α Sj w);jρ ij, i S j w)) := F d rwj ),R Sj, i S w);j j w) 1 ρ 2 ij with S j w) = {i : w i > w j } For any w 1 < w 2 < < w d, we have a 0 w) = α 2d;1 ρ i1, 2 i d)w 1 + α 3d;2 ρ i2, 3 i d)w w d ) a MO w) = β 1 w 1 + β 2 + β 12 )w β S w d Thus, we need to find β S s such that S:d S β 1 = α 2d;1 ρ i1, 2 i d), β 2 + β 12 = α 3d;2 ρ i2, 3 i d),, S:d S β S = 1 By exchanging any two indexes, we obtain a system of equations for β S s: S:j S,S S c j w) β S = α Sj w);jρ ij, i S j w)), 1 j d, 218) where Sj cw) denote the complement of S jw) To construct non-negative solutions for 218), let Z j 1,, Zj d ) be a random vector jointly having the multivariate Cauchy distribution with correlation matrix 27), and also let γ j i := ρ ij/ 1 ρ 2 ij, 1 i, j d and i j Then α Sj w);jρ ij, i S j w)) = Pr{Z j i γj i, i S jw)} Partition the space A j w) := {z 1,, z d ) : c i γ j i, i S jw)} into the following nonoverlapping subsets: for any S with j S, S Sj cw), A j S w) := {z 1,, z d ) : z i γ j i, i S jw); z k γ j k, k Sc jw) S; z l > γ j l, l S {j}} It is easy see that A j w) = S:j S,S S c j w)a j S w) and the Aj S w) are mutually exclusive Therefore, Pr{Z j 1,, Zj d ) Aj S w)} = Pr{Zj 1,, Zj d ) Aj w)} = α Sj w);jρ ij, i S j w)) S:j S,S Sj cw) 11

13 Let, for any S such that j S and S Sj c w) 1 j d), β S = Pr{Z j 1,, Zj d ) Aj S w)} 219) These non-negative β S s satisfy 218) Hence the exponent a 0 ) of the limiting t-ev copula as ν 0 can be written as the exponent a MO ) of a Marshall-Olkin distribution with parameters 219) Remark 28 The explicit construction 219) can be used to evaluate β S s for the Marshall- Olkin representation of the limiting t-ev copula Note that each β S has S different, but equivalent expressions For example, β 12 in Example 26 has the following two expressions: β 12 = α 3;2 ρ 32 ) α 13;2 ρ 12, ρ 32 ) = α 3;1 ρ 31 ) α 23;1 ρ 21, ρ 31 ) The equivalence of these two expressions can be translated into the following structural symmetry for the trivariate standard Cauchy distribution with correlation matrix R: ) ) ρ 32 ρ F 1 21 ρ + F 1 ρ 2 2,R{2,3};1, ρ ρ 2 31 = F 1 ) ρ 31 1 ρ F 2,R{1,3};2 ρ 12 1 ρ 2 12, ) ρ 32, 1 ρ 2 32 where F 1 ) denote the univariate margin The other symmetrical relations can be similarly obtained Our numerical results show that the multivariate t distributions with df ν 1 do not possess such structural symmetries The limiting t-ev copula as ν 0 takes especially a simple form in the bivariate case Let Y i, i = 1, 2, and Y 12 be independent exponential random variables with EY i ) = α 1, i = 1, 2, and EY 12 ) = 1 α) 1, where α = F 1 ρ/ 1 ρ ) 2 Then the limiting distribution of the t-ev copula C EV u 1, u 2 ) as ν 0 is described by the joint distribution of W 1 = min{y 1, Y 12 }, W 2 = min{y 2, Y 12 }; that is, lim ν 0 CEV e w 1, e w 2 ) = exp{ αw 1 + w 2 )} exp{ 1 α)w 1 w 2 )}, for any non-negative w 1, w 2 3 Sharpness of Compatibility Inequalities for Tail Dependence Let λ ij := b ij 1, 1) denote the lower tail dependence parameter of the i, j) margin C ij of a d-dimensional copula C To compare the range of tail dependence of t copulas t-ev copulas) and the bounds in 14), we utilize the correlation inequality 13) It follows from 12) that for a bivariate t copula with ν df, correlation ρ and tail dependence parameter λ, we have ρ = ν + 1 α)/ν α), α = [ T 1 ν+1 λ/2)] 2 31) 12

14 Thus, the lower and upper bounds of ρ ik as a function of λ ij and λ jk are given by B and B + respectively, where 1 B ± ν + 1 [Tν+1 = λ ij/2)] 2 ) ν + 1 [Tν+1 1 λ jk/2)] 2 ) ± 4ν + 1) Tν+1 1 λ ij/2) Tν+1 1 λ jk/2) ν [Tν+1 1 λ ij/2)] 2 ) ν [Tν+1 1 λ jk/2)] 2 ) Since the tail dependence parameter of the t copula is an increasing ) function of its correlation, the ) lower and upper bounds of λ ik are given by 2T ν+1 ν+1)1 B ) 1+B ) and 2T ν+1 ν+1)1 B + ) 1+B + ) respectively As a limiting case of the t-ev copula when ν, the trivariate Hüsler-Reiss copula with parameters δ ij, δ ik, δ jk, satisfying that 1 ϱ ik;j 1, where ϱ ik;j = δij 2 +δ 2 jk δ 2 ik )/2δ 1 ij δ 1 jk ) are δ 1 ij ± δ 1 jk Since λ ik = 1 2Φδ 1 ik ) the bounds for see 212)) Thus, the bounds for δ 1 ik λ ik are given by 2 2Φ Φ 1 1 λ ij /2) ± Φ 1 1 λ jk /2) ), where Φ ) is the standard normal distribution One can see that the upper bounds of both t and Hüsler-Reiss copula are equal to the upper bound of inequality 14), when λ ij = λ jk Table 1 contains representative results of the bounds of λ ik from both copulas and the bounds of inequality 14) for various values of λ ij, λ jk and ν From the table one can clearly see that the bounds of the t copula are closer than the Hüsler- Reiss copula to the bounds of inequality 14) In fact when ν 0 they are identical, except the case for λ ik + λ jk < 1 where the λ ikl becomes greater than zero Note here that 12) and 31) are still well defined for ν = 0 Hüsler-Reiss t, ν = 5 t, ν = 2 t, ν 0 Bounds in 14) λ ij λ jk λ ikl λ iku λ ikl λ iku λ ikl λ iku λ ikl λ iku λ ikl λ iku Table 1: Bounds λ ikl, λ iku ) of the tail dependence parameter λ ik given λ ij and λ jk for Hüsler- Reiss and t copulas for various df ν, along with bounds obtained by inequality 14) Consider the limit of tail dependence as ν 0 It follows from the Cauchy distribution T 1 x) = π 1 arctanx) + 05 that 12) becomes ) ) 1 ρ 1 + ρ λ = 1 + 2π 1 arctan = 1 2π 1 arccos = 1 π 1 arccosρ), 1 + ρ 2 [ and thus ρ = 2 cos π1 λ)/2 )] 2 ) 1 = cos π1 λ) Given λ ij, λ jk in 0, 1), then ρ ij = cosπ1 λ ij )), ρ jk = cosπ1 λ jk )), implying that ρ iku = ρ ij ρ jk + 1 ρ 2 ij 1 ρ 2 jk = cosπλ jk λ ij )) Then the maximum tail dependence 13

15 possible for λ ik is Similarly, ρ ikl = ρ ij ρ jk dependence possible for λ ik is λ ikl = 1 π 1 arccosρ ikl ) = λ iku = 1 π 1 arccosρ iku ) = 1 λ jk λ ij 1 ρ 2 ij { 1 ρ 2 jk = cosπλ ij + λ jk )) Then the minimum tail 1 λ ij λ jk if λ ij + λ jk < 1, 1 2 λ ij λ jk ) = λ ij + λ jk 1 if λ ij + λ jk 1 The latter case comes from πλ ij + λ jk ) π Therefore, the upper bound of 14) is always reached as ν 0 and the lower bound of 14) can be reached as ν 0 if λ ij + λ jk 1 In the remaining case of λ ij + λ jk < 1, λ ikl ν) = 2T ν+1 ν + 1)1 ρ ikl ) 1 + ρ ikl ) is not decreasing at ν 0; see Figure 1 Also note that for 12), ρ > 0 implies λ > 2T ν+1 ν + 1 ), which approaches 2T 1 1) = 025 as ν 0 For larger ν, ρ > 0 implies λ > ɛν) where ɛν) is near 0 For ν 0, if λ ij, λ jk are small, then ρ ij, ρ jk are negative, but ρ ikl can be quite positive Hence 1 λ ij λ jk is not decreasing as λ ij, λ jk get smaller However, Figure 1 shows that λ ikl ν) in 32) comes close to zero for some ν > 0 In order to show the sharpness of inequality 14) for the case of λ ij + λ jk < 1, we use the Marshall-Olkin distribution, or more generally, certain multivariate min-stable exponential distributions Example 31 Consider a trivariate Marshall-Olkin distribution with exponent 217) and the following parameters, β 1 = 3/4, β 2 = 1/2, β 3 = 1/4, β 12 = 0, β 13 = 1/4, β 23 = 1/2, β 123 = 0 It is easy to verify that all the univariate margins are the standard exponential distribution Using the formulas of tail dependence parameters for Marshall-Olkin distributions see, eg, Li, 2007), we have λ 12 = 0, λ 13 = 1/4, λ 23 = 1/2 Since λ 13 +λ 23 < 1, the lower bound of 14) for λ 12 is achieved Notice that this Marshall-Olkin distribution cannot be used to represent the limit of the t-ev copula as df ν 0, because β 1 > β 2 + β 12, which violates the parameter constraints imposed by 218) see also Example 26) Example 32 Let Z i, Z j, Z k be independent standard exponential random variables Let α i, α j, β i, β j, β k be positive constants such that α i + α j = 1 and β i + β j + β k = 1 Let X i = min{z i /α i, Z j /α j }, X j = min{z i /β i, Z j /β j, Z k /β k }, X k = Z k ) 32) 14

16 λ ikl λ = 01 λ = 02 λ = 03 λ = ν Figure 1: Lower bound λ ikl ) of the tail dependence parameter λ ik for the t copula as a function of ν for various values of λ = λ ij = λ jk λ ij + λ jk < 1) The minimum of each curve is positive but very close to 0 Then X i, X j, X k ) has jointly a multivariate min-stable exponential distribution with standard exponential margins Also X i, X k are pairwise independent, but X i, X j and X j, X k are pairwise dependent Hence λ ij > 0, λ jk > 0 and λ ik = 0 so that The bivariate survival functions are given by G ij x i, x j ) = exp{ α i x i β i x j + α j x i β j x j + β k x j )} =: exp{ a ij x i, x j )} G jk x j, x k ) = exp{ β i x j + β j x j + x k )} =: exp{ a jk x j, x k )}, λ ij = 2 a ij 1, 1) = 2 α i β i α j β j β k 33) λ jk = 2 a jk 1, 1) = 2 β i β j 1 = β k 34) Assume, without loss of generality, that α i < β i, α j > β j From 33) and 34) we have that { λ ij = 1 + β j α j α j = β j + 1 λ ij λ jk = β k β i = 1 λ jk β j 35) 15

17 Thus we obtain the family of solutions in 35) if 0 < α j < 1 0 < β i < 1 β j 0, min{λ ij, 1 λ jk } ) 0 < β j < 1 If we select β j 0, min{λ ij, 1 λ jk } ), then we have a family of distributions with λ ij + λ jk < 1 but λ ik = 0 The lower bound of 14) is achieved for this family of distributions 4 Concluding Remarks In this paper the sharpness of inequality among three tail dependence parameters of a trivariate margin) is proved The key is the t copula as ν 0 which shows that the trivariate t copulas cover a wide range of tail dependence This motivated the derivation of the limiting extreme value distribution for the d-variate t copula with ν df; this limit involves the d 1)-dimensional t distribution with ν + 1 df Software exists for the numerical computations of the multivariate t distribution function, for example, Genz and Bretz 2002) and the mvtnorm package in R A conjecture is that for d 4, the set of tail dependence parameters {λ jk : 1 j < k d} for the t and the related copulas might reach the full range Sharpness results in dimensions d 4, for sets of 6 or more bivariate tail dependence parameters, would be hard to try to establish, but based on the sharpness for d = 3 or sets of 3 tail dependence parameters), the range of {λ jk : 1 j < k d} for d-variate t copulas can be compared against if one wants to check if another particular copula family has a wide and flexible range of tail dependence References Demarta, S and McNeil, A J 2005) The t copula and related copulas International Statistical Review, 73: Embrechts, P, McNeil, A J, and Straumann, D 2002) Correlation and dependency in risk management: properties and pitfalls In Risk Management: Value at Risk and Beyond ed M Dempster), pp Cambridge University Press, Cambridge, UK Genz, A and Bretz, F 2002), Methods for the computation of multivariate t-probabilities Journal of Computational and Graphical Statistics, 11: Hüsler, J and Reiss, R-D 1989) Maxima of normal random vectors: between independence and complete dependence Statistics & Probability Letters, 74): Joe, H 1997) Multivariate Models and Dependence Concepts Chapman & Hall, London Kotz, S and Nadarajah, S 2004) Multivariate t Distributions and Their Applications Cambridge University Press, Cambridge, UK 16

18 Kurowicka, D and Cooke, R 2006) Uncertainty Analysis with High Dimensional Dependence Modelling Wiley Series in Probability and Statistics Li, H 2007) Tail dependence comparison of survival Marshall-Olkin copulas To appear in Methodology and Computing in Applied Probability Marshall, A and Olkin, I 1967a) A generalized bivariate exponential distribution Journal of Applied Probability, 4: Marshall, A and Olkin, I 1967b) A multivariate exponential distribution Journal of the American Statistical Association, 62:30 44 McNeil, A J, Frey, R, Embrechts, P 2005) Quantitative Risk Management: Concepts, Techniques, and Tools Princeton University Press, Princeton, New Jersey Wilson, E B 1912) Advanced Calculus Ginn and Company, Boston 17