Extreme value distributions for dependent jointly l n,p -symmetrically distributed random variables

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1 Extreme value distributions for deendent jointly l n, -symmetrically distributed random variables K. Müller and W.-D. Richter, University of Rostock, Institute of Mathematics, Ulmenstraße 69, Haus 3, 857 Rostock, Germany Running headline: Extremes of deendent variables Received: ABSTRACT: A measure-of-cone reresentation of skewed continuous l n, - symmetric distributions, n N, >, is roved following the geometric aroach known for ellitically contoured distributions. On this basis, distributions of extreme values of n deendent random variables are derived if the latter follow a joint continuous l n, -symmetric distribution. Light, heavy, and extremely far tails as well as tail indices are discussed, and new arameters of multivariate tail behavior are introduced. Key words: measure-of-cone reresentation, -generalized Lalace and Gaussian distributions, skewed l n, -symmetric distribution, tail index, light/ heavy center of distribution Introduction Extreme value statistics are of interest not only in robability theory and mathematical statistics, but also in many fields of natural sciences and technique. While Fortin and Clusel 5 resent numerous alications of extreme value statistics to hysics, Shibata 994, Majumdar and Kraivsky, and Castillo deal with alication to corrosion, comuter science, and engineering, resectively. Furthermore, the aearance of extreme value statistics in finance, insurance and actuarial science is dealt with for instance in Embrechts, Klüelberg, and Mikosch 997 and Reiss and Thomas 997. For an alication to reliability theory, see Müller and Richter 5b. General introductions into and surveys over the theory and ractice of extreme value distributions are resented, among other, in David and Nagaraja 3, Leadbetter, Lindgren, and Rootzén 983, Galambos 987, Pfeifer 989, Reiss 989, Reiss, Haßmann, and Thomas 994, and Galambos, Lechner, and Simiu 994.

2 Distributions of extreme value statistics of indeendent and identically distributed random variables rvs are already determined in Gumbel 958. The case of correlated rvs with a joint normal distribution is dealt with, i.a., in Guta and Pillai 965, Nagaraja 98 and Kella 986. The robability density functions dfs of the maximum statistic and of linear combinations of order statistics of arbitrary absolutely continuous deendent rvs, with an emhasis on ellitically contoured samle distributions, are considered in Arellano-Valle and Genton 8 and Arellano-Valle and Genton 7. These considerations are followed u in Jamalizadeh and Balakrishnan and some aer referred to there, and further develoed by reresenting the results with the hel of skewed distributions. Earlier results dealing with this relationshi can be found in Loerfido where the two-dimensional Gaussian case is considered. A geometric aroach to bivariate and multivariate skewed ellitically contoured distributions is resented in Günzel, Richter, Scheutzow, Schicker, and Venz and Richter and Venz 4, resectively, where the measure-of-cone reresentation of these distributions is worked out. In Müller and Richter 5a, some stes of the develoment of the theory of l n, -symmetric distributions and their alications are reviewed. An emhasis of this overview is on geometric measure reresentations and on a methodological study of their alications to the derivation of exact statistical distributions if samles follow a joint continuous l n, -symmetric distribution. Such exact results extend those valid for multivariate sherically symmetric samle distributions. Notice that results on exact extreme value distributions holding if the samle distribution is an arbitrary element of the larger class of ellitically contoured distributions may be again further extended assuming a - generalized ellitically contoured samle distribution. The latter distributions aeared already in Section 3.5 in Arellano-Valle and Richter and were studied from the oint of view of star-shaed distributions in Richter 4 and from that of convex and radially concave contoured distributions in Richter 5a. All these distribution classes rovide more flexibility in modeling data. For more details, we refer to Section. Skewed l n, -symmetric distributions are introduced in Arellano-Valle and Richter and alied to the maximum distribution of continuous l, -symmetrically distributed random vectors in Batún-Cutz, González-Farías, and Richter 3. From a certain oint of view, the aim of the resent aer is to extend this result to the finite number n of deendent rvs following a joint continuous l n, -symmetric distribution. To this end, a geometric aroach to skewed l n, -symmetric distributions, following and generalizing main ideas of the measure-of-cone reresentations from Richter and Venz 4 in the resent situation, will be develoed and, afterwards, used for the derivation of extreme value distributions. From another oint of view, our resent results extend,

3 although in a slightly different form, those derived in Müller and Richter 5a for three or four deendent random variables to the case of an arbitrary finite number of deendent rvs following a joint continuous l n, -symmetric distribution. As an additional result of the resent aer it becomes obvious that the exlicit reresentations of distributions of extreme value statistics derived in the two earlier aers of the authors may be considered as reresentations of skewed distributions being alternatives to the known ones. In order to reresent maximum distributions in terms of skewed distributions, there are two basic aroaches. On the one hand, the authors of Batún-Cutz et al. 3 resent a full-length roof of transforming the result on the maximum df from Müller and Richter 5b directly into the language of skewed distributions. To this end, they start from the Lalace and Gaussian cases, resectively, and extend the results, assing the two-dimensional -ower exonential case, stewise in quick succession to the l, - symmetric case with an arbitrary density generator dg. On the other hand, the authors of Günzel et al. and Richter and Venz 4 deduce a certain measure-of-cone reresentation of skewed ellitically contoured distributions. If one comares the numerous alications of geometric measure reresentation done so far in the literature, one may distinguish between the direct and more advanced alications. To roughly define these terms, direct alications deal with immediate calculations of the so called intersection roortion function of a articular random event under consideration, and more advanced alications deal with tyes of intersection roortion functions being tyical for whole classes of random events. While the small samle studies in the earlier aers of the authors belong to the direct tye of alications of the geometric measure reresentation, the resent aer deals with a more advanced one in Section 5.. The rest of the resent aer is organized as follows. In Section, general information about the considered class of l n, -symmetric distributions and the corresonding class of skewed distributions are given. Based on the l n, -symmetric model assumtion, in Section 3, the cumulative distribution function cdf and the df of extreme value statistics for a finite number of deendent rvs are considered. The density of the maximum is grahically illustrated for several choices of dgs of the l 3, -symmetric samle distributions and for certain values of the shae/ tail arameter >. Studying the asymtotic behavior of the df of the maximum statistic of the comonents of an n-dimensional -generalized Gaussian distributed random vector is another aim of Section 3. In Section 4., the figures of the maximum df of three rvs following a joint l 3, -symmetric Pearson Tye VII or Kotz tye distribution are discussed w.r.t. the heaviness of their tails. The tail index and two new arameters describing the tail behavior of some l n, -symmetric distributions, and the centers of l 3,5 -symmetric Kotz tye distributions are considered 3

4 in Sections 4. and 4.3, resectively. In Section 5, first, a geometric measure-of-cone reresentation of skewed l n, -symmetric distributions is introduced leading afterwards to an advanced geometric method of roof. Second, based uon this, the results of Section 3 are roved and the advanced geometric method of roof is concisely comared to the direct one from Müller and Richter 5a. In Section 6, some conclusions are drawn from the results of the resent aer. Aendix A rovides density generators of and some more basic facts on subclasses of l n, -symmetric distributions. Aendix B deals with the influence which arameters of density generators have onto the heaviness or lightness of multivariate distribution tails. Preliminaries In this section, the model class of l n, -symmetric distributions and the class of skewed l n, - symmetric distributions are introduced and some of their basic roerties are recalled. n Throughout this aer, let > be arbitrary but fixed and let x = x k, x = x,..., x n T R n, denote the -functional which is a norm if and, according to Moszyńska and Richter, an antinorm if,. A function g :,, satisfying the assumtion < Ig < is called a density generating function dgf of an n-variate distribution where Ig = r n gr dr. An random vector X : Ω R n defined on a robability sace Ω,A, P and having the df k= f X x = g x ω n, Ig, x Rn, is said to follow the continuous l n, -symmetric distribution with dgf g where ω n, = Γ n n Γ n denotes the l n,-generalized surface content of the l n, -unit shere S n, = {x R n : x = }, see Richter 9. Further, a dgf g of a continuous l n, -symmetric distribution meeting the condition Ig = ω n, is called dg of this distribution, and denoted by g n. We denote the cdf and the df of the corresonding distribution by Φ g n, and ϕ g,, resectively, where n ϕ g n,x = g n x, x R n. For examles of dgs, for references to the literature on l n, -symmetric distributions, as well as for a discussion of a further asect of notation, we refer to Müller and Richter 5a and the Aendix A, resectively. Remark. According to Richter 9, an l n, -symmetrically distributed random vector 4

5 X with dg g n satisfies the stochastic reresentation X d = R U n 3 where U n is n-dimensional -generalized uniformly distributed on the l n, -unit shere S n,, R and U n are stochastically indeendent and R is a nonnegative random variable with df f R r = ω n, r n g n r, r >. 4 Here and in what follows X d = Z and X Ψ means that the random vectors X and Z follow the same distribution law and that the random vector X follows the distribution law Φ, resectively. Moreover, let I n be the n n unit matrix and n the zero vector in R n. Remark. The density of the nonnegative random variable R is a g n, -generalization of the χ -density, f χ g n, y = ω n, y n g n y y n g n y = ρ n g n, y >. ρ dρ Having in mind the slight change of notation from Richter 4 and Müller and Richter 5a, resectively, this distribution law was defined for arbitrary dgf g in Richter 9 having df f χ g,y = y n gy r n gr dr = y n gy, y >. ρ n gρ dρ and was considered earlier in Richter 99, 7 for the articular cases = and P E, resectively. Remark 3. Let X Φ g,. According to Arellano-Valle and Richter, EX = n n if ER is finite and CovX = EXX T = σ I,g n n if ER is finite where σ =,g n τ n, ER is called the univariate variance comonent and τ n, = Γ Γ 3 n Γ Γ n+. Consequently, the comonents of X are uncorrelated. The comonents of X are only indeendent if P E, see Remark 5 and Aendix A. As announced in Section, now, we further discuss the connection between l n, - symmetric and standardized multivariate ellitically contoured or sherical distributions. As it is well known according to Cambanis, Huang, and Simons 98, the Euclidean stochastic reresentation of an n-dimensional with arameter vector µ R n 5

6 and nonnegative definite matrix Σ R n n having rank k ellitically contoured random vector Z is given by Z d = µ + R U k A 5 where U k is uniformly distributed on the unit shere inr k,r is indeendent of U k, Σ = A T A is a rank factorization of Σ, and the cdf F of R is connected to the characteristic generator φ of Z by φu = Ω k r u dfr, u, where Ω k denoted the characteristic function of U k. In contrast to 5, a non-euclidean stochastic reresentation of Z is given for regular Σ by Z d = µ + R U 6 where R and U are indeendent, R follows a g-generalized χ -distribution which is defined in Richter 99, and U follows the ellisoidal or -generalized uniform robability distribution introduced in Definition 3. in Richter 3. A generalization a of 6 is given in Section 4.7 in Richter 4 where -generalized ellitically contoured distributions are considered. Definition 8 in Section 4.4 of the same aer deals with a stochastic reresentation of even more general star-shaed distributions. It is well known that some ellitical distributions can be obtained as normal variance mixtures. For the general case, see, e.g., McNeil, Frey, and Embrechts 5, and for the articular case of the multivariate skew-t distribution, see Demarta and McNeil 5. In Arslan and Genç 3, a scale mixture exression of exonential ower distributions is given for a generalized-t distribution defined in McDonald and Newey 988. general scale mixture of -generalized normal distributions is dealt with in Section 3.3 in Arellano-Valle and Richter including the secial case of -generalized Student-t distribution. For getting a first imression of the heaviness of the multivariate tails of l n, -symmetric distributions one may study tyical values of the quantile function Q g n, : [, ] R w.r.t. the domain B n, = { x R n : x } defined by A Q g n,q = inf { r > : Φ g n, r B n, q }, q [, ]. 7 According to 3 and Remark, Q g n,q = F R q, q [, ]. where F R has the density 4. In Aendix B, the quantiles Q g n,q are comuted for several values of n, different dgs and q {.9,.95,.99,.995,.999}. According to the work of Loerfido, Jamalizadeh and Balakrishnan, Batún-Cutz et al. 3, and other authors, distributions of extreme value statistics 6

7 are intrinsically connected with certain skewed versions derived from the considered samle distributions. Skewed versions of l n, -symmetric distributions are studied in Arellano-Valle and Richter. To follow these authors, let X = X T, X T T be a random vector having a continuous l k+m, -symmetric distribution with dg g k+m where X : Ω R k and X : Ω R m. We recall that, differing from, the density of X was reresented in Arellano-Valle and Richter as g k+m x. Taking, here and later, this minor change of notation into account, the dg g k k+m the dg of the marginal distribution of X in R k satisfies g k k+m z = ω m, z y z m g k+m y dy, z,. Furthermore, for Λ R m k, Γ = Λ, I m, and Σ = ΓΓ T = I m + ΛΛ T, the cdf of ΓX will be denoted by F m, m x; Σ, g k+m, x R m. Moreover, for every x R k, the conditional density of X given X = x is g k+m x + x g k k+m x = g m [ x ] x, x R m, and the corresonding distribution law is Φ g m this distribution, Y Φ g m [ x ] F m, m x; g [ x ] = R m + [ x ],, then its cdf is,. Let Y be a random vector following g m [ x ] x u du, x R m. A k-dimensional random vector Z having a df of the form f Z z = gk F m, m ; Σ, g m k+m k+m z F m m, Λz; g [ z ], z R k, 8 is said to follow the skewed l k, -symmetric distribution SS k,m, Λ, g k+m with dimensionality arameter m, dg g k+m and skewness/ shae matrix-arameter Λ. Further, the arameter k is called the co-dimensionality arameter and the cdf of Z is denoted by F k,m, ; Λ, g k+m. Notice that F m, x; g m [ x ] = v<x g m v [ x ] dv, and that F m, m ; Σ, g k+m m = m if Σ is diagonal. The following remark deals with the effects of interchanging columns or rows in the matrix-arameter Λ and is roven in Section 5.. Remark 4. a Let M be a k k ermutation matrix and Z SS k,m, Λ, g k+m. Then M Z SS k,m, ΛM T, g k+m where ΛM T arises from Λ by interchanging 7

8 columns. b Let M be a m m ermutation matrix. Then for z R k, F k,m, z; M Λ, g k+m = F k,m, z; Λ, g k+m, i.e. SS k,m, Λ, g k+m = SS k,m, M Λ, g k+m where M Λ arises from Λ by interchanging rows. According to Arellano-Valle and Richter, skewed l k, -symmetric distributions are constructed via selection mechanisms from l n, -symmetric distributions. Particularly, if X : Ω R k and X : Ω R m are again two subvectors of a random vector X, X Φ g k+m,, then L X X < ΛX = SS k,m, Λ, g k+m 9 where LY denotes the distribution law of the random vector Y. Therefore, art b of Remark 4 reflects the exchangeability of the comonents of an l n, -symmetrically distributed random vector within the skewed l k, -symmetrical distributions. 3 Extreme value distributions for arbitrary finite samle sizes We recall that secific results for exact distributions of order statistics of u to three and extreme value statistics u to four deendent rvs following a joint continuous l n, - symmetric distribution, n {, 3, 4}, are roved in earlier aers of the authors by directly alying the geometric measure reresentation of l n, -symmetric distributions. In Section 3., exact extreme value distributions are derived if an arbitrary finite number of deendent rvs follows a continuous l n, -symmetric distribution. To this end, an advanced geometric method of roof will be develoed in Section 5 following main ideas for geometrically reresenting skewed ellitically contoured distributions in Richter and Venz 4. In Section 3., our results are grahically visualized. On the one hand, figures of densities are drawn for the secial case of jointly trivariate 3-generalized Gaussian distributed rvs and, on the other hand, for the case of three deendent rvs following a joint l 3, -symmetric Kotz tye and a joint l 3, -symmetric Pearson Tye VII distribution, resectively. Another aim of this section is to rovide an idea of the asymtotic behavior of the maximum df for n-variate -generalized Gaussian samle distribution as n tends to infinity. 8

9 3. Dimension and co-dimension reresentations The results of this section reflect strong connections between skewed distributions and distributions of extremes. Such tye of connection can already be seen in Loerfido, Jamalizadeh and Balakrishnan, and in aers of several other authors. The resent results are derived based uon the geometric measure reresentation in Richter 9. This reresentation alies directly if only small numbers of deendent rvs are considered. In Batún-Cutz et al. 3, the articular result on the maximum df of two deendent rvs being jointly l, -symmetrically distributed is transformed directly into the tyical reresentation of skewed distributions. Here, we use the geometric reresentation of l n, -symmetric measures in a more advanced way. To be more concrete, we derive from it a measure-of-cone reresentation of skewed l k, -symmetric distributions in Corollary, Section 5. Let E ν denote the ν n ν matrix whose st column is ν =,..., T R ν and whose remaining n ν columns are ν-dimensional zero vectors. Theorem. If X Φ g n,, for every ν {,..., n }, the cdf F n:n of the maximum statistic of the comonents of X satisfies the reresentation F n:n t = ν + F ν, ν ; Σ, g n ν Fn ν,ν, tn ν ; E ν, g n, t R, where Σ = I ν + E ν E νt = I ν + ν T ν. For arbitrary n N and >, Theorem rovides numerous reresentations of the cdf of the maximum statistic from l n, -symmetrically distributed oulations in terms of skewed distributions. In articular, these are alternatives to that given in Theorem in Müller and Richter 5b for n = and Theorems and 3 in Müller and Richter 5a for n = 3 and n = 4, resectively. In the case of n =, the equivalence of these two alternative reresentations is shown in Batún-Cutz et al. 3 by direct integral transformation. Furthermore, in the secific case = and ν = n, the result of Theorem is covered by Theorem 7 in Jamalizadeh and Balakrishnan. Now, we briefly discuss the imact of the arameter ν {,..., n } in Theorem. Recalling that F n ν,ν, tn ν ; E ν, g n is the cdf of SS n ν,ν, E ν, g n and considering the construction of skewed l n ν, -symmetric distributions via select mechanisms again, X and X in equation 9 are real-valued n ν- and ν-dimensional subvectors of an l n, -symmetrically with dg g n distributed random vector X, resectively. Hence, the arameter ν defines the dimensionality of the conditioning subvector in equation 9 and, imlicitly, the dimension of the skewed l k, -symmetric distribution which is used to reresent the maximum cdf in Theorem. Esecially for ν =, E = e n T n where e denotes the jth unit vector of R n, and the matrix j 9

10 I + T = is diagonal. Thus, F, ; I + T, g = = and equation ν+ reads as F n:n t = F n,, t n ; e n T, g n, t R. Therefore, maximum distributions for continuous l n, -symmetric vectors may articularly be reresented as skewed l n, -symmetric distributions. For all the other arameters ν {,..., n }, the matrix I ν + ν T ν is not diagonal. Due to this, the normalizing constant F ν, ν ; I ν + ν T ν, g ν n is not as easy to handle with as in the case ν = and, in general, its exact value is unknown. Nevertheless, all reresentations of the maximum cdf given in are well treatable since the corresonding maximum dfs, see Corollary, do not deend on the mentioned normalizing constant. Vice versa, it may be sometimes of interest to read equation in a reverse order, meaning that for every ν {,..., n }, the cdf of Z, Z SS n ν,ν, E ν, g n, at the articular argument t n ν, t R, satisfies F n ν,ν, tn ν ; E ν, g n = ν + F ν, ν ; Σ, g ν n F n:n t, t R, where Σ = I ν + ν T ν and F n:n denotes the cdf of the maximum statistic of the comonents of X, X Φ g n,. Using the direct alication of the geometric measure reresentation shortly discussed in Section, the cdf F n:n is already determined for n {, 3, 4} in earlier aers of the authors. Thus, by substituting F n:n on the right hand side of equation for n {, 3, 4} by these revious reresentations, one gets alternative reresentations of the cdf F n ν,ν, tn ν ; E ν i, g n, t R, to that following from Section as the comonentwise defined integral of the df of SS n ν,ν, E ν i, g n over the region {z R n ν : z < t n ν }. Summarizing this section u to here, there are n equivalent ossibilities of reresenting the maximum cdf F n:n using skewed distributions. This effect also occurs in the reresentation of the maximum df f n:n as it can be seen in the following corollary being roved in Section 5.. Corollary. Let X Φ g n,. For every t R and ν {,..., n }, f n:n t = ν + n ν + ν + z Dt z Dt g n ν n g n ν n t + z F where Dt = {z R n ν : z < t n ν }. Moreover, f n:n t = n g n t F n, t + z F ν, ν, z ν ; g ν [ t + z ] t ν ; g ν [ t + z ] dz dz tn ; g n [ t ], t R.

11 Using the general relation between maximum and minimum statistics if the samle distribution is symmetric and continuous, and the functions rovided by Theorem and Corollary, the minimum cdf and df of the comonents of X, X Φ g n,, are given by F :n t = F n:n t and f :n t = f n:n t, resectively. In articular, for ν = n, this yields f :n t = n g t F n, tn ; g n [ t ], t R. 3 In due consideration of the mentioned slight variation of notation for l n, -symmetric densities, formula equals for n = the result in Batún-Cutz et al. 3. It is worthwhile to note that the structure of our results on extremes of several deendent variables in and 3 is similar to that of the corresonding results of the df of the maximum and the minimum statistic, resectively, of n indeendent and identically distributed rvs, see David and Nagaraja 3. Remark 5. Let X be n-dimensional -generalized Gaussian distributed, X Φ n g with P E, g n P Er = n Γ ex Further, let ϕ t = g P E t, t R, and Φ t = { r t }, r >. ϕ s ds, t R, denote the df and the cdf of the one-dimensional marginal distribution, resectively. The dfs and 3 of the extreme value statistics of the comonents of X simlify to f n:n t = n ϕ t Φ t n, t R, and f :n t = n ϕ t Φ t n, t R, resectively. Note that, in these secific cases, our reresentations and 3 also follow from David and Nagaraja 3 since the comonents of -generalized Gaussian distributed random vectors are indeendent. 3. Visualization of the maximum density In the resent section, the df of the maximum statistic of deendent, jointly l n, - symmetrically distributed rvs is illustrated for some choices of the dg g n and the shae/ tail arameter >. Figures -3 show the maximum df of the comonents of a

12 three-dimensional -generalized Gaussian distributed random vector, i.e. choosing the dg g 3 P E, accomanied by histogram lots of samles of sizes from 3 u to.5 5 for {,, 3}. These Figures do not only demonstrate the numerical correctness of our evaluations but it also indicate in a certain rough sense how large samle sizes should be when simulating extreme value densities under non-standard model assumtions. Figures 4 and 5 visualize the df of the maximum statistic of the comonents of l 3, - generalized Kotz tye and Pearson Tye VII, resectively, distributed random vectors for different values of the arameter, >. The definitions of the dgs g n Kt;M,β,γ and g n P T 7;M,ν of these subclasses of the l n, -symmetric distributions are given in the Aendix A. Note that the resent choice of the shae/ tail arameter and of the arameters aearing in the definitions of the dgs coincides with that in Figures and 3 in Müller and Richter 5a. Thus, one can comare the grahs of the df of the maximum statistics resented here with that of the median statistic drawn there. Another aim of this section is to give a visual imression of the asymtotic behavior of the df of the maximum statistic for increasing samle sizes. This will be done in n-dimensional -generalized normally distributed oulations, i.e. in the case of indeendent comonents. Using the dg g n P E and the four choices of the arameter as in Figures 4 and 5, the imact of an increasing samle size onto the shae of the maximum df is reflected in Figure 6. Furthermore, in this figure, one can erceive the imact of arameter which is, on the one hand, a shae arameter and, on the other hand, a tail arameter since the shae of the multivariate density level sets deends on it and the tail df of the underlying random vector becomes lighter if increases, resectively. Note that the axes in Figures 4, 5 and 6 are scaled differently, and that both the left and the right hand sides of Figures 4a, 5a, 5b and 6a show a black grah as a resective benchmark. Moreover, note that illustrations of the minimum df can be received if the grahs of the corresonding maximum dfs are mirrored at the ordinate axis. 4 Tails and centers of maximum distributions In this section, we review some of the figures of Section 3. in detail. Moreover, we give additional information on the tail index and on heaviness and on lightness of tails of l n, -symmetric distributions. 4. Light, heavy and extremely far tails The influence that the arameter ν > of an l 3, -symmetric Pearson Tye VII distribution with arameter M = 3 has onto the heaviness of the tails of the median distribution

13 a samle size b samle size c samle size d samle size.5 5 Figure : Maximum df f 3:3 and histogram for =, increasing samle sizes and dg g 3 P E a samle size b samle size c samle size d samle size.5 5 Figure : Maximum df f 3:3 and histogram for =, increasing samle sizes and dg g 3 P E. 3

14 a samle size b samle size c samle size d samle size.5 5 Figure 3: Maximum df f 3:3 and histogram for = 3, increasing samle sizes and dg g 3 P E. 4 M=,β=,γ= M=,β=,γ=5 M=,β=,γ= M=,β=5,γ= M=,β=,γ= 3.5 M=,β=,γ= M=,β=,γ= M=5,β=,γ= M=,β=,γ= a = M=,β=,γ= M=,β=,γ= M=,β=,γ=5 M=,β=,γ= M=,β=5,γ= M=,β=,γ= M=5,β=,γ= M=,β=,γ=.5 M=,β=/,γ= M=,β=,γ= M=,β=,γ=5 M=,β=,γ= M=,β=5,γ= M=,β=,γ= M=5,β=,γ= M=,β=,γ= b = c = 4

15 .5 M=,β=/3,γ= M=,β=,γ= M=,β=,γ=5 M=,β=,γ= M=,β=5,γ= M=,β=,γ= M=5,β=,γ= M=,β=,γ= d = 3 Figure 4: Maximum df f 3:3 for {,,, 3}, dg g 3 Kt;M,β,γ, and several choices of the arameters M > 3, β >, and γ > black dashed: the secial case of trivariate -generalized Gaussian distribution M=6.5,ν= M=8.5,ν= M=,ν= M=8.5,ν= M=,ν= M=6.5,ν= M=6.5,ν= M=6.5,ν= a =.5 M=3.5,ν= M=5.5,ν= M=7.5,ν= M=5.5,ν= M=7.5,ν= M=3.5,ν= M=3.5,ν= M=3.5,ν= b = M=,ν= M=4,ν= M=6,ν= M=,ν= M=,ν=3 M=4,ν= M=6,ν=3.5 M=.5,ν= M=3.5,ν= M=5.5,ν= M=.5,ν= M=.5,ν=3 M=3.5,ν= M=5.5,ν= c = d = 3 Figure 5: Maximum df f 3:3 for {,,, 3}, dg g 3 P T 7;M,ν, and several choices of the arameters M > 3 and ν >. 5

16 of three deendent rvs is discussed in Section 4 of Müller and Richter 5a. In the resent section, first, an analog study for the case of the maximum distribution in such oulations is done. Second, we examine the heaviness of the tails of the maximum distribution from other l 3, -symmetric samle distributions with resect to their arameters. n= n=3.9 n=4.8 n=5 n= n= n= n=5 n= 4 n=5* a = n= n=3 n=4 n=5 n= n= n=5 n= 4 n=5* 4 b = n= n=3 n=4 n=5 n= n= n=5 n= 4 n=5* 4 n= 5 c = 6

17 n= n=3 n=4 n=5 n= n= n=5 n= 4 n=5* 4 n= 5 d = 3 Figure 6: Maximum df f n:n for different n and dg g n P E if {,,, 3}. Note that the grahs of the right hand side of Figure 5a convey the imression that the visualized densities build a monotonically decreasing sequence of functions. The more detailed views in Figure 7, however, show the regions of intersection between the black and the green solid, the black and the green dashed, and the green solid and the green dashed grahs, being aroximately { 8; 5}, { ; 8}, and { 3; 53}, resectively M=6.5,ν= M=6.5,ν= M=6.5,ν= M=6.5,ν= M=6.5,ν= M=6.5,ν= a b M=6.5,ν= M=6.5,ν= M=6.5,ν= M=6.5,ν= 6.6 M=6.5,ν= M=6.5,ν= c d Figure 7: Some detailed views on the grahs of the right hand side of Figure 5a. Correcting a otentially misleading imression coming from considering just the restricted central art of the densities in Figure 5 thus makes it necessary to study the 7

18 far tails of the same distributions. Particularly, Figure 7 emhasizes the increasing heaviness of tails of the distribution of the maximum statistic of three deendent rvs following a joint l 3, -symmetric distribution with dg g 3 if ν increases. The same P T 7; 3,ν tendency can be seen in the case of l 3, -symmetric Pearson Tye VII samle distribution with constant arameter M = 7 and increasing arameter ν. In Table, the integral A M,νz, z = z f 3:3 t dt of the maximum df f 3:3 over some asymmetric intervals [z, z ] z is numerically comuted for l 3, -symmetric Pearson Tye VII distributed oulations with arameters, M {, } 3,, 7 and ν {,, 3} to get a more detailed numerical imression of the tendency of heaviness of tails. A M,νz, z = M = 3 = M = 7 z z ν = ν = ν = ν = ν = ν = Table : Heaviness of the extremely far reaching tails of maximum distribution in jointly { l 3, -symmetrically Pearson Tye VII distributed oulations with, M, } 3,, 7 and ν {,, 3}. For the sake of comarison of heaviness of the tails of the median and the maximum distribution, our resent consideration is for the class of l 3, -symmetric Pearson Tye VII distributions with the same different values of ν > and M = 3 as in Section 4 in Müller and Richter 5a. Equally, interreting the values in Table, one can concentrate on one of Figures 5c and 5d and check the same effect of increase of heaviness of tails if the arameter M > 3 is constant and the arameter ν > increases. According to Table, all the three cases in Figure 5a cover only a small art of the entire robability mass. Therefore, the behavior of the grahs outside the considered interval [.5;.5] is not clearly redictable. However, for the case of ν = 3, Figure 8 shows the df of the maximum statistic of rvs following a joint l 3, -symmetric distribution with dg g 3 over the interval [ ; ], suggesting a monotonically increasing P T 7; 3,3 behavior over the negative real line and a monotonically decreasing one over the ositive real line. Note that only an extremely small roortion of robability mass generates a eak of the density function close right to the zero oint and that the overwhelming art of robability mass is seemingly uniformly distributed on an extremely long interval. Additionally, in Table, the integral A M,z, z is numerically evaluated for = and M {, 4, 6} and = 3 and M { 3, 7, }, resectively and several asymmetric real intervals [z, z ] where " " denotes the case that the value of A M,z, z rounded 8

19 Figure 8: Maximum df f 3:3 for dg g 3 P T 7; 3,3 over the interval [ ; ]. to the sixth decimal lace equals. These values emhasize a decreasing heaviness of the tails of the maximum distribution in jointly l 3, -symmetrically Pearson Tye VII distributed samles if the arameter M > 3 constant. increases and the arameter ν > is A z M,z, z = z 4 = z = 3 z 4 = z = 3 z = z = 3 z = z = 3 z = z = 5 z = 9 z = 6 z = 9 M = = M = M = = 3 M = M = M = Table : Heaviness of the tails of maximum distribution in jointly l 3, - symmetrically { Pearson Tye VII } distributed oulations with, M,,, 4,, 6, 3, 3, 3, 7, 3, and ν =. Finally, Table 3, shows different heaviness of the tails of the distribution of the maximum statistic of three jointly l 3, -symmetric Kotz tye distributed rvs for the choices of arameters from Figure 4b by numerically comuting the integral B M,β,γ z, z = z z f 3:3 t dt of the maximum df in such oulations. This suggests that the heaviness of the tails of the maximum distribution decreases if either γ or β increases and increases if M increases where, in each of the cases, the other two arameters are constant. 4. Tail indices While the tail of the distribution of the univariate maximum statistic in jointly l 3, - symmetrically Pearson Tye VII or Kotz tye distributed samles is exlored in Section 9

20 B M,β,γ z, z z =. z =.5 z =. z =.35 z =.5 z = z =.5 z =.75 z = z =.5 z = z = 3.5 M =, β =, γ = M =, β =, γ = M =, β =, γ = M =, β =, γ = M =, β = 5, γ = M =, β =, γ = M = 5, β =, γ = M =, β =, γ = Table 3: Heaviness of the tails of maximum distribution in jointly l 3, -symmetrically Kotz tye distributed oulations for = and the choices of arameters M > 3, β > and γ > from Figure 4b. 4., here, we consider the tail of the multivariate l n, -symmetric distribution itself. To this end, we restrict our considerations to Pearson Tye VII, Pearson Tye II, and Kotz tye distributions, see Aendix A. We determine the tail index of regularly varying distributions in Section 4.. and study the multivariate tail behavior for light tails and bounded suorts in Sections , resectively. 4.. Heavy Tails Let us call a random variable regularly varying with tail index α > w.r.t. the - functional, >, if there exist a ositive constant α and a robability law S on the Borel-σ-field B n S n, of subsets of the -unit shere such that for every x > x α µ z ; S as z where the symbol means weak convergence, and µ z M; = P X X > xz, M X P X > z, M B n S n,. Furthermore, S is called the sectral measure w.r.t.. Note that this comlies with the common notion of a regularly varying distribution w.r.t. the -norm if. Now, let X Φ g n, with dg g n. Thus, because of the stochastic reresentation 3, P R > xz, U n M µ z M; =, M B n S n,, P R > z

21 with R and U n being indeendent. If P R > xz lim z P R > z = x α, x >, i.e., according to the notion in Resnick 987, if the survival function of the univariate random variable R is regularly varying at with index α, then the n-dimensional -generalized uniform distribution on S n, is the sectral measure of Φ g n, w.r.t., SM = P U n M, M B n S n,. Examle. In the articular case P T 7;M,ν with M > n of L Hôital s rule yields P R > xz lim z P R > z = lim z xz z r n g n P T 7;M,νr dr r n g n P T 7;M,νr dr and ν >, the alication = x M n, x >. Consequently, the l n, -symmetric Pearson Tye VII distribution with arameters M > n and ν > has tail index M n. In the case n = and, this result is already covered by Examle 6 in Richter 5b. Remark 6. If Kt;M,β,γ with M > n, β > and γ >, then P R > xz lim z P R > z = xn+m z lim e βxγ z γ, < x < =, x =., x > Hence, the tail index of l n, -symmetric Kotz tye distribution does not exist. 4.. Light Tails Adoting de Haan s notion of Γ-variation, see Resnick 987 and original references cited therein, let us call a random vector Γ-varying w.r.t. the -functional, >, if there exist a ositive function f and a robability law S on B n S n, such that for every x R e x µ Γ z ; S as z where µ Γ z M; = P X X > z + xfz, M X P X > z, M B n S n,.

22 Here, f is called an auxiliary function. If X Φ g n, then µ Γ z M; = P R > z + xfz P R > z P U n M and hence S is the -generalized uniform distribution on the Borel-σ-field over S n, if P R > z + xfz lim z P R > z = e x, x R, i.e. the survival function of the univariate -radius variable R of X is Γ-varying with auxiliary function f. Examle. We consider the secial case of an l n, -symmetric Kotz tye distribution with arameters M > n, β > and γ >, i.e. gn = g n Kt;M,β,γ, and denote the df and the cdf of the Radius variable R by f Kt and F Kt, resectively. For fz =, z >, βγ z γ and for every x R, from of the asymtotic equivalence relations F Kt r f Ktr βγ r γ as r and lim z β z + xfz γ + βz γ = x, it follows that F Kt z + xfz lim z F Kt z = e x. Thus, a random vector following an l n, -symmetric Kotz tye distribution with arameters M > n, β > and γ > is Γ-varying w.r.t. the -functional and with auxiliary function fz = z γ, z >. βγ Remark 7. For a random vector being l n, -symmetrically Pearson Tye II distributed with arameter ν > one can verify neither the roerty of regular variation nor that of Γ-variation w.r.t. the -functional, >, since the suort of df of the corresonding radius variable is bounded Bounded suorts Let X be an l n, -symmetrically contoured random vector such that the distribution of X has a bounded suort, and let us denote the right endoint of if by x E. If there exist a ositive constant α and a robability law S on B n S n, such that for every x > x α µ b z ; S as z

23 where µ b z M; = P X > x E, X M xz X P, M B n S X > x E n,, z then we call the random vector X bounded regularly varying with tail index α > w.r.t., >. In this case, µ b zγ ; P R > x E xz = P P U R > x E n z and S is the n-dimensional -generalized uniform measure on S n, if for every x > lim z P R > x E xz P = x α, R > x E z i.e. if the survival function of the univariate random variable x E R at with index α. is regularly varying Examle 3. In the secial case P T ;ν with ν >, on the one hand, x E = and, on the other hand, P R > xz lim z P = R > x lim n xz ν xz z z z = x ν+ z for all x >. Therefore, the l n, -symmetric Pearson Tye II distribution with arameter ν > or a random vector following that distribution is bounded regularly varying with tail index ν + w.r.t Light and heavy distribution centers While extremely long concentration intervals and extremely far tails of robability distributions were studied in Sections 4. and 4., here the focus is on the centers of l 3,5 - symmetric Kotz tye distributions for certain choices of arameters. Nevertheless, it is worthwhile to mention that l n, -symmetric Kotz tye distributions have relatively light tails caused by the exonential art of their dgs. The monomial art of the dg ensures that the heaviness of the distribution center can be modeled with the hel of the arameter M. It can be seen from Figure 9a, that the choice of M = is the decisive factor to have a heavy distribution center. Standard examles of this distributional tye are ower exonential and, articularly, Gaussian and Lalace distributions, and their l n, -generalizations. Additionally, in the case M =, Figure 9a shows that the arameter β > controls mainly the height and the arameter γ > 3

24 mainly the decay behavior of the dg. In the case M =, the arameter β as well as γ regulate the height of the dg. Furthermore, they induce a shift of the robability mass. The effect of these choices of arameters on the shae of the df of the maximum distribution of three deendent rvs following a joint l 3,5 -symmetric Kotz tye distribution can be seen in Figure 9b. Finally, one can comare Figure 9b with Figure 4 to get an imression of the imact of a further increase of the arameter. 5 Proofs 5. Measure-of-cone reresentations of skewed l k, -symmetric distributions An initial ste of the roof of Theorem deals with deriving a reresentation of the cdfs of skewed l k, -symmetric distributions with dimensionality arameter m in terms of secific l n, -symmetric measures of cones by analogy to what was done in Richter and Venz 4 for skewed ellitically contoured distributions with dimensionality arameter m =. Afterwards, this measure-of-cone reresentation is used to rove Remark 4. Let a random vector Z follow the skewed l k, -symmetric distribution with dimensionality arameter m, dg g k+m, and matrix-arameter Λ R m k, and let e n j, j =,..., n, still denote the jth standard unit vector of R n. For arbitrary z R k, we consider the cone A z = C m a,,..., a,m, a,..., a k ; z m { = x R k+m : a T,lx < } k { } x R k+m : a T i x < z i l= i= M=,β=,γ= M=,β=,γ=3 M=,β=,γ=5 M=,β=3,γ= M=,β=5,γ= M=,β=,γ= M=,β=,γ=5 M=,β=,γ= M=,β=5,γ= M=,β=,γ= M=5,β=,γ= M=,β=,γ= a The dg g 3 Kt;M,β,γ for certain arameters M > 3 5, β > and γ >. 4

25 M=,β=,γ= M=,β=,γ=3 M=,β=,γ=5 M=,β=3,γ= M=,β=5,γ= M=,β=,γ= M=,β=,γ=5 M=,β=,γ= M=,β=5,γ= M=,β=,γ= M=5,β=,γ= M=,β=,γ= b The df of the maximum statistic for the same arameters as in Figure 9a. Figure 9: The l 3,5 -symmetric Kotz tye samle distribution. where the quantities a,l = Γ T e m l, l =,..., m, a i = e k+m i, i =,..., k, and Γ = Λ, I m are as in Section. This cone generalizes that in Richter and Venz 4 where the case m = is dealt with. Note that A z has its vertex at z T, Λz T T. Furthermore, A z is the intersection of k + m half saces, m of whom containing the origin in its boundary. Lemma A secific measure-of-cone reresentation of the cdf F k,m, ; Λ, g k+m. If Z SS k,m, Λ, g k+m, then the cdf of Z allows the reresentation F k,m, z; Λ, g k+m = F m, m ; I m + ΛΛ T, g m k+m Φ g k+m,a z, z R k, where F m, ; Im + ΛΛ T, g k+m m is the normalizing constant from 8. Proof. Let C m, = F m, m;i m+λλ T,g m k+m F k,m, z; Λ, g k+m = C m, where the integral z = C m, = C m, z z z R m +. Then g k k+m ζ F m m, Λζ; g [ ζ ] dζ g k k+m ζ R m + g m [ ζ ] Λζ ξ dξ dζ g k+m ζ + Λζ ξ dξ dζ, z R k, hζ dζ, z R k, is defined as a k-fold one. Further, using the notation X = X, X T where the vectors X and X take their values in R k and 5

26 R m, resectively, F k,m, z; Λ, g k+m = C m, z R m + ϕ g k+m, ζ, Λζ ξ dξ dζ = C m, PX < z, X < ΛX, z R k. Here, ϕ g k+m, is the df of X, and { x, x T R k+m : x < z, x < Λx } { = x = x, x } T R k+m : x < z, Γx < { = x R k+m : e k+m T i x < zi, i =,..., k, Γ T e m } T l x <, l =,..., m = A z. Thus, F k,m, z; Λ, g k+m = C m, PX < z, X < ΛX = C m, PX A z = C m, Φ g k+m,a z, z R k. Note that Theorem in Richter and Venz 4 follows from Lemma for m = and =. Let O denote the l n, -generalized surface content on B n S n, and F :, [, the l n, -shere intersection-roortion function if defined by r F A, r = O r A S n, O S n, = O r A S n, ω n,, A B n. 4 According to Richter 9, but with suitably adated notations as in Richter 4, 5a and Müller and Richter 5a, for arbitrary > and n N, the continuous l n, -symmetric distribution Φ g n, with dg g n satisfies the geometric measure reresentation Φ g n, A = I F A, r r n g n r dr, A B n. 5 g n Note that this formula was first roved for = and P E in Richter 985, and for = and arbitrary dgf g in Richter 99. Using formula 5 and the symmetry of both O and the l n, -unit shere S n,, the following lemma rovides some invariance roerties of the l n, -symmetric robability 6

27 measure. Lemma. The l n, -symmetric robability measure Φ g n, is ermutation and sign invariant. Proof. Let σ : {,..., n} {,..., n} be a ermutation defined by i σi, i =,..., n, and M the corresonding ermutation matrix, i.e. M = e n σ en σ σn en. Since ermutation matrices are orthogonal matrices, M is invertible and M = M T. Further, let X Φ g n,. By the geometric measure reresentation, PMX A = Φ g n,m T A = I F M T A, r r n g n r dr, A B n, g n where, because of M T S n, = S n, and O M T A S n, r = O M T A S n, r = O A S n, r, the if satisfies F M T A, r = F A, r, A B n. Thus, PMX A = Φ g n,a = PX A, A B n. Let D be an n n sign matrix, i.e. D = diag{d,..., d n } with d i {, }, i =,..., n. D is an orthogonal matrix and, by analogous considerations as before, PDX A = PX DA = Φ g n,da = Φ g n,a = PX A, A B n. Consequently, Φ g n, is a member of the class SI of sign invariant distributions, considered in Arellano-Valle and del Pino 4. Note that one can rove Lemma alternatively without using formula 5, starting from Φ g,a = ϕ n g,x dx = n A g n x dx and using the invariance roerties of the -functional. A Next, we are going to generalize the secific measure-of-cone reresentation formula for the cdf of a skewed l k, -symmetrically distributed random vector given in Lemma. Doing this, first, a class of cones is introduced and, then, the ermutation and sign invariance of the measure Φ g k+m, is basically utilized. We recall that the invariance of Φ g n, with resect to all orthogonal transformations was used to construct general geometric measure reresentations for n = in Günzel et al. and for arbitrary n in Richter and Venz 4. Let Λ: R k R m be a matrix with entries Λ j,i, j =,..., m, i =,..., k. Further, let I = {i,..., i k } {,..., k + m} denote a set of indices with I = k and D a k + m k + m sign matrix. Moreover, let V I,D Λ; z = D v,..., v k+m T with v il = z l for l =,..., k and v jν = Λz ν = k Λ ν,l z l for j ν {,..., k + m}\i and 7 l=

28 ν =,..., m. For every z R k, we define a cone with vertex at V I,D Λ; z by C I,D Λ; z = {Dx R k+m : x il < z l, l =,..., k k x jν < Λ ν,l x il, j ν {,..., k + m}\i, ν =,..., m}. l= From here on, the collection of such cones will be denoted by C k,m Λ; z. Particularly, A z = C {,...,k},ik+m Λ; z is an element from the class C k,m Λ; z of cones. Consequently, C k,m Λ; z = {MDA z: M Π k+m, D S k+m } where Π n and S n denote the sets of n n ermutation and sign matrices, resectively. Corollary A general measure-of-cone reresentation of the cdf F k,m, ; Λ, g k+m. The skewed distribution in Lemma allows each of the reresentations F k,m, z; Λ, g k+m = F m, m ; I m + ΛΛ T, g m k+m Φ g k+m, C{i,...,i k },D Λ; z, z R k, where C {i,...,i k },DΛ; z is an arbitrary element of the class C k,m Λ; z. Proof. By definition of C k,m Λ; z, there is a ermutation matrix M such that C {i,...,i k },DΛ; z = M C {,...,k},d Λ; z = M D A z. Then M defines a ermutation σ with σl = i l, l =,..., k, and σk+,..., σk+m {,..., k + m}\{i,..., i k } so that the bijectivity of σ is ensured. By Lemma and Lemma, and with C m, = F m, m;i m+λλ T,g m k+m as in the roof of Lemma, F k,m, z; Λ, g k+m = C m, Φ g k+m, A z = C m, Φ g k+m, M D A z = C m, Φ g k+m, C{i,...,i k },D Λ; z, z R k. Proof of Remark 4. Initializing the roof of art a, let M be a k k ermutation matrix and Z SS k,m, Λ, g k+m. Since M ξ = ξ, ξ R k, and detm =, PM Z < z = PZ < M T z = = F m, M T z f Z ζ dζ = m ; I m + Λ Λ T, g m k+m 8 z z f Z M T ξ dξ g k k+m ξ F m, Λ ξ; g m [ ξ ] dξ,

29 z R k, where Λ = ΛM T. Thus, M Z SS k,m, ΛM T, g k+m. To start the roof of art b, let M be a m m ermutation matrix in R m and Z SS k,m, M Λ, g k+m. As in the roof of Lemma, but with X = X, X T Φ g k+m,, F k,m, z; M Λ, g k+m = F m, PX < z, X < M ΛX m ; I m + M ΛM Λ T, g m k+m, z R k. Note that { x, x T R k+m : x < z, x < M Λx } { x =, x } T R k+m : x < z, M T x < Λx { x =, M x } T R k+m : x < z, x < Λx = M A z with M = diag [I k, M ] R k+m k+m being a ermutation matrix. The set A z = M A z is an element of the class C k,m Λ; z. By Lemma, PX < z, X < M ΛX = PX A z = PX A z. Additionally, let Γ = Λ, I m and Γ = M Λ, I m, then = F m, m ; I m + M ΛM Λ T, g k+m m = PΓ X < m g k+m x dx } {x=x,x T R k+m : M Λx <x = } {y=y,y T R k+m : Λy <y g k+m M y dy = PΓX < m = F m, m ; I m + ΛΛ T, g k+m m. Finally, F k,m, z; M Λ, g k+m = = F m, F m, m ; I m + M ΛM Λ T, g m k+m m ; I m + ΛΛ T, g m k+m = F k,m, z; Λ, g k+m, z R k. PX A z PX A z 9

30 5. Alying the advanced geometric method to extremes We recall that advanced alications of the geometric measure reresentation 5 make use of tyes of intersection ercentage functions 4 being valid for whole classes of random events. The classes of events considered here are cones generated by intersecting half saces. In this section, the measure-of-cone reresentations from Corollary are mainly used to rove Theorem which, in turn, is basic to establish Corollary and Remark 5. Moreover, the direct and the advanced geometric methods considered in this aer are briefly comared. Let A n nt = { x,..., x n T R n : x < t,..., x n < t }, t R, be a sublevel set generated by the maximum statistic of an n-dimensional random vector X = X,..., X n T. Then, P max{x,..., X n } < t = P X A n nt, t R. Moreover, illustrations of the set A n nt may be found in the two earlier aers of the authors for n {, 3}. Proof of Theorem. Let ν {,..., n } be fixed. Further, let B,t = {x R : x < t, x < x } and B,t = {x R : x < t, x x }. Then A t = B B,t,t is a disjoint decomosition of the cone A t with vertex at t, t T, see Figure. Similarly, on the one hand, one has that A 3 3t = B t B,t where Figure : Disjoint decomosition of A t for t >. B t = {x < t, x < t, x 3 < x } and B,t = {x < t, x 3 < t, x x 3 }. On the other hand, using this and the decomosition of A t again, A 3 3t = {x < t, x < t, x 3 < x } {x < t, x 3 < t, x x 3 } = {x < t, x < x, x 3 < x } {x < t, x 3 < x, x x 3 } = {x < t, x, x 3 T A x } B,t B,t B,t B,t 3