Exact distributions of order statistics of dependent random variables from l n,p -symmetric sample distributions,

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1 Exact distibutions of ode statistics of deendent andom vaiables fom l n, -symmetic samle distibutions, n {3, 4} K. Mülle ) and W.-D. Richte 2) ), 2) Univesity of Rostock, Institute of Mathematics, Ulmenstaße 69, Haus 3, 857 Rostock, Gemany Running headline: Ode statistics of deendent vaiables Received: ABSTRACT: Integal eesentations of the exact distibutions of ode statistics ae deived in a geometic way when thee o fou andom vaiables deend fom each othe as the comonents of continuous l n, -symmetically distibuted andom vectos do, n {3, 4}, >. Once the eesentations ae imlemented in a comute ogam, it is easy to change the density geneato of the l n, -symmetic distibution with anothe one fo newly evaluating the distibution of inteest. Key wods: density geneato, exteme value statistics, geometic measue eesentation, -genealized Gaussian and Lalace distibutions, heavy tails Intoduction It is well known that uncoelatedness of a finite numbe of andom vaiables (vs) imlies thei indeendence if thei joint multidimensional distibution is a Gaussian one. Moe secifically, if the density geneating function (dgf) of a sheically distibuted andom vecto is that of a Gaussian vecto then the comonents of this vecto ae indeendent vs. Fo any othe choice of the dgf, these vs deend fom each othe in a cetain way. Similaly, if a andom vecto follows a continuous l n, -symmetic o l n, -sheical distibution, >, its n comonents ae indeendent if its dgf is that of a suitably defined n-dimensional -owe exonential distibution, and only in this case. Theefoe, studying distibutions of functions of sheically o l n, -symmetically distibuted andom vectos means, in geneal, studying distibutions unde secific deendence assumtions w..t. the joint samling distibution. Note that the class of

2 l n,2 -symmetic distibutions is just that of all sheical distibutions. The tye of deendence among the comonents of a continuous l n, -symmetically distibuted andom vecto deends on both the dgf and the aamete. One might call, fo shot, this deendence the l n, -symmety deendence. Ode statistics ae useful tools in aametic and nonaametic statistics as well as, e.g., in eliability theoy and othe alied eseach aeas. The distibutions of ode statistics of indeendent and identically distibuted andom vaiables ae exhaustively dealt with in the last decades, see e.g. David and Nagaaja (23). The obability density function (df) of the maximum statistic as well as that of a linea combination of ode statistics of abitay absolutely continuous deendent andom vaiables is studied in Aellano-Valle and Genton (28) and Aellano-Valle and Genton (27), esectively. In both aes, secial emhasis is on the case that the joint multivaiate samle distibution is an ellitically contoued distibution. In Jamalizadeh and Balakishnan (2) and some aes efeed to thee, the latte investigations ae followed u and futhe develoed by eesenting the esults with the hel of skewed distibutions. Fo a elated esult fo continuous l 2, -symmetically distibuted samle vectos, see Batún-Cutz, González-Faías, and Richte (23). The class of ellitically contoued distibutions extends that of sheical distibutions, see Fang, Kotz, and Ng (99). Anothe extension is the class of l n, -symmetic o l n, - sheical distibutions. This class has been intoduced in Osiewalski and Steel (993), and dealt with late on, e.g., in Guta and Song (997). A geometic measue eesentation of these distibutions was oved in Richte (29), see equation (3) in Section 5.. This eesentation found alications to simulation in Kalke and Richte (23) and to the deivation of cetain exact distibutions in Kalke, Richte, and Thaue (23). In Mülle and Richte (25), integal eesentations of exact distibutions of exteme value statistics of l 2, -symmetically distibuted samles ae oved making ossible to easily change a given density geneato with anothe one. Hee, we extend these esults to dimensions thee and fou. The aim of the esent ae, howeve, is twice. On the one hand, as indicated, we contibute new esults on the exact distibution of ode statistics of thee o fou deendent vs if the samle distibution is an l n, -symmetic one, n {3, 4}. On the othe hand, we conduce to a systematic study of cases in which the geometic measue eesentation successfully alies. While the fist aim of this ae needs no futhe exlanation, the second one will to be discussed a bit close in the following. In Richte (24a), a oblem is dealt with which was consideed befoehand by seveal authos in a seies of aes and by using diffeent methods. Reoving cetain of the known esults with a new method, alying the geometic measue eesentation, actually needs most of the sace in Richte (24a). Fo the subsequent ste of 2

3 substantially extending the class of andom vaiables ossessing the same oety of inteest, howeve, only little additional effot is needed. This way, a sometimes involved method suddenly mutates to a oweful ancillay tool of mathematical wok. Fo a geneal discussion on the value of eoving, see Silveman (994). Anothe effect of systematically alying a geometic measue eesentation is discoveed in Dietich, Kalke, and Richte (23). Among othe things, the authos develo a new integal eesentation of the cumulative distibution function (cdf) of the lagest eigenvalue of a cetain Wishat distibuted andom matix although anothe eesentation in tems of hyegeometic functions has been well established aleady in the liteatue fo a long time. The non-anticiated wage fo these methodological effots was in numeical stability oeties of the new esult. Futhemoe, the systematic geometic measue theoetical studies in Günzel, Richte, Scheutzow, Schicke, and Venz (22) and Richte and Venz (24) bing moe stuctue into a vaiety of well-known oofs and esults on skewed distibutions, and noticeable genealize seveal well established esults. In all these cases, new esults ae oved fo vs deending fom each othe unde the influence of a dgf and ossibly additional aametes. In ode to summaize the two main aims of this ae, besides oving new esults on distibutions of ode statistics, we extend the ange whee geometic measue eesentations successfully aly. This way, we contibute to establish such eesentations as standad ancillay tools of actical wok in obability theoy and statistics. The est of the esent ae is oganized as follows. In Section 2, geneal infomation on the model class of l n, -symmetic distibutions ae given. Assuming this l n, -symmetic model class, in Section 3, ou main esults on the cdf and df of maximum, median, and minimum statistics of thee deendent vs and on the cdf of exteme value statistics of fou deendent vs ae esented. The df of the median is visualized, one the one hand, fo tivaiate -genealized Gaussian distibuted oulations, = 3, jointly with histogam lots fo inceasing samle sizes and, on the othe hand, fo l 3, -symmetically Kotz tye and Peason Tye VII distibuted oulations fo seveal choices of aametes. Section 4 is aimed to discuss some figues given befoehand and to inteelate the undelying distibutions with heavy tailed ones. In Section 5, the esults of Section 3 ae oved. In aticula, basics of the geometic method of oof ae exlained in Section 5.. In the final Section 6, some conclusions ae dawn fom the esults in the esent ae. 3

4 2 The model class We conside the model class of continuous l n, -symmetic distibutions in this ae as a subclass of the class of sta-shaed distibutions. This oint of view leads to a slight change of notation fo continuous l n, -symmetic distibutions, comaed with evious aes dealing with these distibutions. Let K R n be a sta body having the oigin in its inteio and let S denote its toological bounday. The functional h K : R n [, ) defined by h K (x) = inf{λ > : x λk}, x R n, is known as the Minkowski functional of K whee λk = {(λx,..., λx n ) T : (x,..., x n ) T K}. A function g : (, ) (, ) satisfying the assumtion < I(g) < is called density geneating function (dgf) of an n-vaiate distibution whee I(g) = n g() d. Accoding to Richte (24b), moeove assuming the homogeneity of degee one and a cetain smoothness of h K, a obability measue having the df ϕ g,k (x) = C(g, K)g(h K (x)), x R n, is called a sta-shaed distibution with density contou defining sta body K, and denoted by Φ g,k. The nomalizing constant allows the eesentation C(g, K) = O S (S) I(g) whee O S denotes the sta genealized suface measue on S and is defined as well in Richte (24b). If K is the unit ball of the finite-dimensional nomed o antinomed sace (R n, ) then h K (x) = x, and Φ g,k is called a nom o antinom contoued distibution in R n, esectively, see Richte (25b) fo the 2-dimensional and Richte (25a) fo the geneal case. Fo the notion of an antinom, we efe to Moszyńska and Richte (22). Thoughout this ae, let >. We denote the l n, -unit ball and the l n, -unit shee by K n, = {x R n : x } and S n, = {x R n : x = }, esectively, ( n ) whee x = x k, x = (x,..., x n ) T R n, stands fo the -functional. Then, k= h Kn, (x) = x, and h Kn, is a nom if and, accoding to Moszyńska and Richte (22), an antinom if (, ). Futhe, the sta genealized suface measue O Sn, matches with the l n, -genealized suface measue O defined in Richte (29), and ω n, denotes the l n, -genealized suface content of S n,, ω n, = (2Γ( )) n n Γ( n ). In aticula, an n-dimensional andom vecto X : Ω R n defined on a obability sace (Ω,A, P) and having a df f X (x) = ϕ g,kn, (x) = g( x ), x ω n, I(g) Rn, is said to follow the continuous l n, -symmetic distibution Φ g, with dgf g. Fo shot, the density f X = ϕ g,kn, is witten as f X = ϕ g,. This df is nom contoued if and adially 4

5 concave sta-shaed if (, ). Fom now on, we assume that g is secifically chosen as a density geneato (dg), i.e. the nomalizing constant meets the condition ω n, I(g) =. In othe wods, g is chosen in such a way that ϕ g, (x) = g( x ), x R n. This notation of an l n, -symmetic df slightly diffes fom the notation f X (x) = g( x ), x R n, used in Guta and Song (997), Richte (29), Aellano-Valle and Richte (22), Batún-Cutz et al. (23), Kalke et al. (23), Mülle and Richte (25), as well as Fang et al. (99) and Günzel et al. (22) in the sheical case. Because of g(c) = g(c ), c >, we obtain I(g) = I n, g, whee the notation I n, g, = n g( ) d is used in evious aes. The emaining at of this section deals with examles of density geneatos of continuous l n, -symmetic distibutions. In slightly othe notation, these and othe examles can be found aleady in Guta and Song (997), and fo the case = 2 in Fang et al. (99). Note that only g = g P E in Examle 3 geneates indeendence of the comonents of the andom vecto. Examle. The dg of the l n, -symmetic (o n-dimensional -genealized) Kotz tye distibution with aametes β, γ > and M > n is g Kt;M,β,γ () = n n+(m ) γβ γ Γ ( ) n 2Γ ( ) Γ ( ) (M ) ex{ β γ }, >. n+(m ) γ If = 2, aying attention to the change of notation, this is the dg of standadized Kotz tye distibution, see Nadaajah (23). In Guta and Song (997), Φ gkt;m,β,γ, has aamete N = M, and is called -genealized Weibull distibution. Examle 2. The dg of the l n, -symmetic owe exonential distibution with aamete γ > is g P E;γ () = n γ Γ ( ) n { } 2Γ ( ) ( ) n γ Γ n ex γ, >, γ i.e. g P E;γ = g Kt;,/,γ. If = 2, this dg geneates the standadized multivaiate owe exonential distibution, see Gómez, Gómez-Villegas, and Maín (998), whose univaiate fom is intoduced in Subbotin (923). Unde vaious aameteizations, and sometimes called exonential owe distibution, the univaiate distibution Φ gp E;γ,2 is studied in Box and Tiao (973), Osiewalski and Steel (993), and Nadaajah (25, 26). Examle 3. The aticula function g P E; = g P E is called the dg of the n-dimensional 5

6 -owe exonential o -genealized Gaussian o -genealized Lalace distibution, g P E () = n ) 2Γ ( ex { }, >. If = o = 2, g P E is the dg of the n-dimensional Lalace o Gaussian distibution, esectively. Examle 4. The dg of the l n, -symmetic Peason Tye VII distibution with aametes ν > and M > n is g P T 7;M,ν () = n 2Γ ( ) Γ(M) ν n Γ ( M n ) ) M ( +, >. ν Examle 5. The dg g St;ν of the l n, -symmetic Student-t distibution with ν > degees of feedom is defined as g St;ν () = n Γ( n+ν 2Γ ( ) ) ( ) ν n Γ ν In addition, g St;ν = g P T 7; n+ν,ν. ) n+ν ( +, >. ν Examle 6. The dg g C of the l n, -symmetic Cauchy distibution satisfies g C = g St; as it is well known in the sheical case = 2. if t A Let A (t) = denote the indicato function of the set A. othewise Examle 7. The dg of the l n, -symmetic Peason Tye II distibution with aamete ν > is ( g P T 2;ν () = n Γ n 2Γ ( ) + ν + ) ( ) ν (,) (), >. Γ(ν + ) 3 Exact distibutions of ode statistics Fo the est of the ae, we assume that the andom vaiables X,..., X n ae the comonents of the andom vecto X, X Φ g,, fo an abitay shae/ tail aamete 6

7 > as well as an abitay dg g. Futhemoe, we denote the coesonding vecto of ode statistics by X (od) (n) = (X :n,..., X n:n ) T and the cdf and df of X k:n, k =,..., n, by F k:n and f k:n, esectively. The following esult descibes the basic stuctue of ou eesentations of F k:n and f k:n. Lemma (Seaating oety). The cdf and the df of X k:n allow the eesentations F k:n (t) = f k:n (t) = f(t, ) n g() d, () h(t, ) n g() d, (2) t R, with functions f, h: R (, ) R not deending on the dg g. Fo simlicity of notation, we do not indicate that the functions f and h deend on the integes n and k, and on the aamete. Note that the influence of g onto the distibution of X k:n is seaated in Lemma fom that of all the othe aametes. Once the functions f and h ae imlemented in a comute ogam, it is easy to change a cetain dg g with anothe one fo newly evaluating the functions F k:n and f k:n. Figues -3 show the median density fo diffeent tyes of the dg (ecognize diffeent scaling in diffeent ictues). The undelying esults of Sections 3. and 3.2 secify the functions f and h, and will be deived on using the geometic measue eesentation (3), see Section Maximum, median, and minimum distibutions of thee deendent vs In ode to define the function f in () fo n = 3 in a dense fom, we will make use of the following notations. Fo any eals ρ a < ρ b fom [, ) and any functions ϕ a and ϕ b maing [, ) to [, 2π) and satisfying ϕ a (ρ) < ϕ b (ρ) fo all ρ (, ), let whee G (2) H(ρ a, ρ b ; ϕ a, ϕ b ) = (ϕ a (ρ), ϕ b (ρ)) = ϕ b(ρ) ϕ a(ρ) ρ b ρ a ρ ( ρ ) G (2) (ϕ a (ρ), ϕ b (ρ)) dρ dϕ (N (ϕ)) 2 with N (ϕ) = ( cos (ϕ) + sin (ϕ) ) denotes the -genealized unifom distibution on S 2, (ρ), see Richte (28a, 28b). Hee, S n, (ρ) = ρ S n, = {x R n : x = ρ} denotes the l n, -shee with -adius ρ 7

8 ( ) t (, ). Futhemoe, fo all (, ) and t R, let α t, (ρ) = actan and ρ t H i (ρ a, ρ b ) = H(ρ a, ρ b ; ϕ a,i, ϕ b,i ), i =,..., 5, whee the functions ϕ a,i and ϕ b,i ae given in Table. i = i = 2 i = 3 i = 4 i = 5 π ϕ a,i (ρ) π + α t, (ρ) π α t, (ρ) α π 2 t,(ρ) π α t, (ρ) + α 2 t,(ρ) 3π ϕ b,i (ρ) α π 3π 2 t,(ρ) α 2 t, (ρ) + α 3π 2 t,(ρ) α 2 t,(ρ) Table : Definitions of the functions ϕ a,i (ρ) and ϕ b,i (ρ), i =,..., 5, ρ >. Theoem. The cdf of the maximum statistic satisfies eesentation () with n = k = 3 and 2 t f(t, ) = (,] (t) ( 3 t, ) ()H (, t ) + (, ) (t) (,t) ()ω 3, ( + [t, 2t) () ω 3, ) 2 2 ( t ) 2 8H 2 ( t, ) + [ 2t, 3t) () t ω 2 3, H 3 ( 2t t, ) + H 4( t, ) + H 3 ( t, 2t ) + H 4( t, ) + [ 3t, ) () t ω 2 3, 2 + H 4( t 2, ) + H 3( t, ) + H 3 ( t, 2t ). In ode to deive fom this esult a tightly looking eesentation of the function h in (2), fo any ρ a < ρ b fom (, ) and ψ a < ψ b fom [, 2π), we use the notations and k ρ (ρ a, ρ b ; ϕ a, ϕ b ) = ρ b ρ a ρ ( ρ ) α t,(ρ) [ N 2 k ψ (ψ a, ψ b ) = 3 ( t ) 2 G (2) (ψ a, ψ b ) (ϕ a (ρ)) + N 2 (ϕ b (ρ)) ] dρ, ( ) t β t, = α t, t = actan 2 t whee k ψ and β t, ae defined fo all (, ) and t R. Note that α t,(ρ) = d dt α t,(ρ) = (ρ t ) + t (ρ t ) t 2 + (ρ t ) 2. 8

9 Finally, we ut k ρ,i (ρ a, ρ b ) = k ρ (ρ a, ρ b ; ϕ a,i, ϕ b,i ) fo i =,..., 5 and k ρ,6 (ρ a, ρ b ) = k ρ (ρ a, ρ b ; ϕ a,2, ϕ a,2 ). Coollay. The df of the maximum statistic satisfies eesentation (2) with n = k = 3 and h(t, ) = (,] (t) ( 3 t, ) () k ρ, ( + (, ) (t) [t, 2t) () t 2 k ψ ( π 2 β t,, β t, ) + k ρ,3 ( 2 t, t ) + k ψ (π + β t,, 3π 2 β t,) ( t ) 2 ω 3, + 4k ρ,6 ( t, ) + [ 2t, 3t) () t, 2t ) + [ 3t, ) () t ω 3, + k ρ,4( t 2, ) + k ψ (π β t,, 3π 2 + β t,) 3 t ( t ) + k ρ,3 ( t, 2t ). 2π() + k ρ,4 ( t, ) It is wothwhile to mention that the cdf and the df of the minimum statistic satisfy the eesentations F :3 (t) = F 3:3 ( t) and f :3 (t) = f 3:3 ( t), t R, esectively. Theoem 2. The cdf of the median statistic satisfies eesentation () with n = 3, k = 2 and f(t, ) = [ 2t, 3t) () 2H 5 ( t, + [ 3t, ) () 2H 5 ( t, 2 t t ) + 2H (, ) t ) + 2H ( t, ), if t, and f(t, ) = ω 3, f( t, ), if t >. Coollay 2. The df of the median statistic satisfies eesentation (2) with n = 3, k = 2 and h(t, ) = [ 2 t, 3 t ) () 2k ρ,5 ( t + 2k ρ, ( + 2k ρ,5 ( t,, t ) + 2k ψ ( π 2 + β t,, 3π 2 β t,) 2 t, ) + [ 3 t, ) () 3 ( t ) π() t ) + 2k ρ, ( t, ). 9

10 Figues -3 give an imession of how the density f 2:3 in eesentation (2) looks like fo diffeent tyes of dgs and seveal diffeent aamete choices. In aticula, Figue deals with the aticula indeendence case of dg g = g P E and = 3 (c.f. Examle 3, fomula (2) and Coollay 2). Note that the numeical coectness of ou evaluations is evealed by adding histogam lots of samles of inceasing sizes fom 3 u to Also illustating the df of the median statistic but now of thee deendent vs following a joint continuous l 3, -symmetic distibution, Figues 2 and 3 deal with geneally deendence geneating dgs of Kotz tye and Peason Tye VII, esectively. Note the diffeent scales of axes of odinates as well as of abscissas, and that Figues 3(a) and 3(b) ae futhe discussed unde seveal asects in Section Exteme value distibutions of fou deendent vs In the esent section, we estict ou consideations to the cdfs of exteme values, i.e. to the functions F k:4 in () with k = 4 and k =, esectively. Fo any ρ a < ρ b fom (, ), any functions ϕ a and ϕ b maing (, ) to [, 2π) and satisfying ϕ a (ρ) < ϕ b (ρ) fo all ρ (, ), and any functions θ a and θ b maing (, ) [, 2π) to [, 2π) and satisfying θ a (ρ, ϕ) < θ b (ρ, ϕ) fo all (ρ, ϕ) (, ) [, 2π), let L(ρ a, ρ b ; ϕ a, ϕ b ; θ a, θ b ) = whee ρ b ρ a ρ 2 ( ρ ) G (3) G (3) (ϕ a (ρ), ϕ b (ρ); θ a (ρ, ϕ a (ρ)), θ b (ρ, ϕ b (ρ))) = (ϕ a (ρ), ϕ b (ρ); θ a (ρ, ϕ a (ρ)), θ b (ρ, ϕ b (ρ))) dρ ϕ b (ρ) θ b (ρ,ϕ b (ρ) ϕ a(ρ) θ a(ρ,ϕ a(ρ)) sin (θ) dθ dϕ N 2 (θ)n 2 (ϕ) denotes the -genealized unifom distibution on S 3, (ρ) with ρ (, ), and sin is the -genealized sine function defined in Richte (27). Moeove, let γ t, (ρ) = actan t ρ 2 t define a aametic function maing [, ) to [, π/2), fo all [, ) and t R. Recognize that γ t, () = β t, fo all t and. Additionally, let δt,(ρ, ϕ) = π 2 +actan ρ sin (ϕ) t ( ) and δ + t t,(ρ, ϕ) = actan cot(ϕ) +

11 (a) samle size 3 (b) samle size (c) samle size 5 (d) samle size Figue : Median df f 2:3 and histogam fo = 3, inceasing samle sizes and dg g = g P E M=,β=2,γ= M=2,β=2,γ=2 M=2,β=2,γ=5 M=2,β=2,γ= M=2,β=5,γ=2 M=2,β=,γ=2 M=5,β=2,γ=2 M=,β=2,γ= M=,β=,γ= M=2,β=2,γ=2 M=2,β=2,γ=5 M=2,β=2,γ= M=2,β=5,γ=2 M=2,β=,γ=2 M=5,β=2,γ=2 M=,β=2,γ= (a) = 2 (b) =.5 M=,β=/2,γ= M=2,β=2,γ=2 M=2,β=2,γ=5 M=2,β=2,γ= M=2,β=5,γ=2 M=2,β=,γ=2 M=5,β=2,γ=2 M=,β=2,γ=2.8.6 M=,β=/3,γ= M=2,β=2,γ=2 M=2,β=2,γ=5 M=2,β=2,γ= M=2,β=5,γ=2 M=2,β=,γ=2 M=5,β=2,γ=2 M=,β=2,γ= (c) = 2 (d) = 3 Figue 2: Median df f 2:3 fo {,, 2, 2 3}, dg g = g Kt;M,β,γ, and seveal choices of the aametes M > 3, β >, and γ >.

12 9 8 7 M=6.5,ν= M=8.5,ν= M=,ν= M=8.5,ν=2 M=,ν= M=6.5,ν= M=6.5,ν=2 M=6.5,ν= (a) = M=3.5,ν= M=5.5,ν= M=7.5,ν= M=5.5,ν=2 M=7.5,ν= M=3.5,ν= M=3.5,ν=2 M=3.5,ν= (b) = M=2,ν= M=4,ν= M=6,ν= M=2,ν=2 M=2,ν=3 M=4,ν=2 M=6,ν= M=.5,ν= M=3.5,ν= M=5.5,ν= M=.5,ν=2 M=.5,ν=3 M=3.5,ν=2 M=5.5,ν= (c) = 2 (d) = 3 Figue 3: Median df f 2:3 fo {,, 2, 2 3}, dg g = g P T 7;M,ν, and seveal choices of the aametes M > 3 and ν >. be aametic functions defined on (, ) [, 2π), and, fo all aametes and t, 3 t L (ϕ a (ρ), ϕ b (ρ)) = L(, t ; ϕ a (ρ), ϕ b (ρ); π 2 + α t,, δt,), L 2, (ϕ a (ρ), ϕ b (ρ)) = L( t, ; ϕ a(ρ), ϕ b (ρ);, π 2 α t,), L 2,2 (ϕ a (ρ), ϕ b (ρ)) = L( t, ; ϕ a(ρ), ϕ b (ρ);, δ t,), + L 3, (ϕ a (ρ), ϕ b (ρ)) = L( t, t ; ϕ a (ρ), ϕ b (ρ);, π 2 α t,), L 3,2 (ϕ a (ρ), ϕ b (ρ)) = L( t, t ; ϕ a (ρ), ϕ b (ρ);, δ t,). + Theoem 3. The cdf of the maximum statistic satisfies eesentation () with n = k = 4 2

13 and f(t, ) = (,] (t) ( 4 t, ) ()2L (π + γ t,, 5π 4 ) + (, )(t) (,t) ()ω 4, + [t, 2t) () ω 4, ω 4, 2 ( 3 t ) 2L2, (, π) + [ 2t, 3t) () ω 4, + 6L 3, (, π) ω ( 3 4, t ) 6L2, ( π 2 2 γ t,, γ t, ) 6L 2, (π γ t,, 3π 2 + γ t,) 2L 2,2 (γ t,, π γ t, ) + [ 3t, 4t) () ω 4, 6L 2, (π γ t,, 3π 2 + γ t,) ω ( 3 4, t ) 2L2,2 ( π 2 4, π γ t,) + 3L 3, (π γ t,, 3π 2 + γ t,) + 3L 3, ( π 2 γ t,, γ t, ) + 6L 3,2 (γ t,, π γ t, ) + [ 4t, ) () ω 4, ω 4, 2 ( 3 t ) 2H2,2 ( π 4, π γ t,) 6H 2, (π γ t,, 3π 2 + γ t,) + 3H 3, (π γ t,, 3π 2 + γ t,) + 6H 3,2 ( π 4, π γ t,). 4 Heavy tails Distibutions having heavy tails lay an imotant ole in statistical actice and find esecially many alications to insuance and financial mathematics. The median df f 2:3 lotted in Figue 3 deals with heavy tails whee the dg of X is of l 3, -symmetic Peason Tye VII which includes both Student and Cauchy tye samle distibutions. It aeas to be tyical in such cases that only vey few obability mass is concentated aound the distibution cente leading on the ight hand sides of Figues 3(a) and 3(b) to the misleading imession that the dawn densities could build a monotonically deceasing sequence of functions. By zooming into the ight hand side of Figues 3(a), howeve, and taking the symmety w..t. axis of odinates into account, one detects the oints of intesection of the black and the geen solid, the black and the geen dashed, and the geen solid and the geen dashed gahs at t /2 ±5, t 3/4 ±23, and t 5/6 ±44, esectively, see Figue 4. A simila exlanation avoids a otential misundestanding in the case of Figue 3(b). Futhemoe, the Figues 3(a) and 4 suggest the visual imession that the tail heaviness of the distibution of the median statistic of thee deendent vs following a joint l 3, -symmetic Peason Tye VII distibution inceases if the aamete M is 2 3

14 x M=6.5,ν= M=6.5,ν=2 M=6.5,ν=3 x 3.7 M=6.5,ν= M=6.5,ν=2.6 M=6.5,ν= (a) (b) Figue 4: Zoom into the ight hand side of Figue 3(a). A ν (z) z =.5 z = z = 3 z = 4 z = 6 ν = ν = ν = Table 2: Inteval obabilities of the samle median in jointly l 3, -symmetically Peason 2 Tye VII distibuted samles with M = 3 and ν {, 2, 3}. 2 constant and the aamete ν > inceases. Fo secific values of inteval obabilities see Table 2 whee the values A ν (z) = z f 2:3 (t) dt ae numeically comuted fo z {.5,, 3, 4, 6 }, shae/ tail aamete = 2, and dg g P T 7;M,ν with aametes M = 3 and ν {, 2, 3}. If M is constant and ν > inceases, such a manne of 2 heaviness of tails can be obseved in all cases of Figue 3. z 5 Poofs In ode to oof the assetions fom Section 3, the geneal method of oof and some basics on alying this method to ode statistics ae esented in Sections 5. and 5.2, esectively. Aftewads, the claimed esults on the distibutions of ode statistics fo thee deendent vs and exteme value statistics fo fou deendent vs ae established wheeas the details of oofs decease in quantity in late ats of these sections. 5. Basics of the geometic method of oof Let T : R n R be any statistic and A(t) = {x R n : T(x) < t} a sublevel set (sls) geneated by it. If X Φ g, with an abitay dg g, the cdf of T(X) is F T (t) = Φ g, (A(t)), t R. 4

15 The geometic measue eesentation in Richte (29), with notations as descibed in Section 2 suitably adated to the ones used in Richte (24b) and hee, alies Φ g, (A(t)) = ω n, F (A(t), ) n g() d (3) whee the l n, -shee intesection-ootion function (if) F : (, ) [, ) is defined on B n by F (A, ) = O ([ ) A] S n,. O (S n, ) Accoding to Richte (29), the l n, -genealized suface content is defined on B n by O (A) = G(A S n,) n j= x j dx + G(A S + n,) n j= x j dx whee G(A) := {(x,..., x n ) K n, : x n R : (x,..., x n ) A} and S +( ) n, = {x S n, : x n ( ) }. Hence, F T (t) = ([ ] ) O A(t) S n, n g() d. Since O ([ A(t)] S n, ) does not deend on the dg g, this eesentation oves the fist at of Lemma, i.e. () and the indeendence of the function f fom the dg g, wheeas T is chosen as an abitay ode statistic. The second at of Lemma follows by the Leibniz integal ule. Now, we eae fo the oofs of the esults fom Sections 3. and 3.2. With the hel of the l n, -sheical coodinate tansfomation SPH (n ) : [, ) [, π) (n 3) [, 2π) R n and its coesonding invese maing, see Richte (27), the cdf of T(X) allows the geneal eesentation F T (t) = M (t,) h(ρ, ϕ) d(ρ, ϕ) + M + (t,) h(ρ, ϕ) d(ρ, ϕ) n g() d (4) whee h(ρ, ϕ) = ( ρ ) M +( ) (t,) J ( ) SPH (n ) (ρ, ϕ), = ( ) ( ([ ] SPH (n ) G A(t) )) S n, +( ), 5

16 and J ( ) n 2 SPH (n ) (ρ, ϕ) = ρ n 2 (sin ϕ i ) n 2 i i= (N (ϕ i )) n i is the Jacobian of SPH (n ). Thus, it emains to detemine the domains of integation M + (t,) and M (t,) fo all cases consideed in Sections 3. and Geneal eesentations of the domains of integation M +( ) (t,) As it can be seen fom Section 5., studying the sets G( [ A(t)] S n,) and G( [ A(t)] S + n,) lays a fundamental ole fo the alication of the geometic measue eesentation fomula. The esent section is aimed to ove geneal eesentations of these sets if the geneating statistic is any ode statistic. Moe secific eesentations will be deived fom it in the next section and will be used thee to ove esults fo all cases consideed in Sections 3. and 3.2. Fo k {,..., n}, let A n k(t) = {x R n : at least k comonents of x ae less than t}, t R, be a sls geneated by the kth ode statistic of an n-dimensional andom vecto. An illustation of the set A n k(t) can be seen in Figue 5 fo (n, k) {(3, 3), (3, 2)} and t <. z z x y x (t,t,t) (t,t,t) y A 3 3(t) A 3 2(t) (a) Sls of the maximum (b) Sls of the median Figue 5: Sls of ode statistics of thee vaiables and t <. Remak. Let X be a continuous and symmetically with esect to the oigin distibuted andom vecto, X X, and let F k:n (t) = P (X k:n < t) be the cdf of the kth ode statistic X k:n of X. Then, fo k =,..., n and evey t R, F n k+,n (t) = F k,n ( t), and F k:n (t) = P (X A n k(t)). 6

17 Lemma 2. If t, then ([ ] ) G An k(t) Sn, = Kn, ([ ] ) G An k(t) S n, + and, if t >, ([ ] ) G An k(t) Sn, ([ ] ) G An k(t) S n, + = = K n, An k (t) t An k (t) [ = K n, An k (t), t R n, [ An k (t) K n, An k K n, An k ] (t), ] (t) whee A m l (t) = if l = o l > m, Å denotes the toological inteio of the set A Rn, K n, (ρ) = {x R n : x ρ} the l n, -ball with -adius ρ (, ), and R n, (ρ a, ρ b ) = K n, (ρ b )\K n, (ρ a ) = {x R n : ρ a < x ρ b } the l n, -laye with -adii ρ a < ρ b. Poof. Let t, B = [ An }, and B2 = [ An k (t) ] R. Then k (t)] { x n R: x n < t An k(t) = B B 2 whee B =, if k =, and B 2 =, if k = n. Note that, fo evey x = (x,..., x n ) T B Sn,, (x,..., x n ) T An k (t), x n < t n, and x i =. Since x n > t, it follows That is why evey element of G(B S n,) = { (x,..., x n ) T n An k (t) and x i t < (x,..., x n ) T R n :! x n : i= n i= i=. x i =, x n < t, (x,..., x n ) T } An k (t) is an element of C := K t n, An k (t). 7

18 In othe wods, G(B Sn,) C. Let (x,..., x n ) T C. Choosing ρ = n x i, it follows ρ < t. Futhe, one can uniquely choose x n = ρ < such that ρ + x n =, i.e. (x,..., x n, x n ) Sn,. Then, x n < t and C G(B Sn,). Hence, G(B Sn,) = K t n, An k (t). Fo any x = (x,..., x n ) T B 2 Sn,, i= (x,..., x n ) T An k (t) and x n. As G(B 2 S n,) = {(x,..., x n ) T R n :! x n : An k (t)}, fo any (x,..., x n ) T G(B 2 S n,) n x i =, (x,..., x n ) T i= (x,..., x n ) T C 2 := K n, An k (t), i.e., G(B 2 Sn,) C 2. Let (x,..., x n ) T C 2. Choosing ρ = n i= x i, it follows ρ. Futhe, let x n = ρ such that ρ + x n =. Because of (x,..., x n ) T [ An k (t) ] C 2, we have (x,..., x n, x n ) T [ R: z } B 2. Thus, C 2 G(B 2 Sn,), and consequently, An k (t) ] {z G(B 2 Sn,) = K n, An k (t). Summaizing the above esults, the fist assetion of the lemma follows. The othe cases can be dealt with in an analogous way. The next ste of analyzing the sets G( [ A(t)] S n,) and G( [ A(t)] S + n,) consists of numeous case studies. Because the numbe of cases inceases if the numbe of vs does, we estict the outline of this way mainly to the case of thee vs. 5.3 Secific eesentations of the domains of integation fo consideing the maximum of thee deendent vs This section demonstates that, in the case of thee deendent vs, the geometic method of oof alies as successful as in Mülle and Richte (25) whee the case of samle size two was dealt with. The esent calculations may also seve as an oientation fo the deivation of analogous esults in moe geneal sta-shaed model classes. Poof of Theoem. To get the exact cdf of the consideed statistic, accoding to equa- 8

19 tion (4), it emains to eesent the sets G ([ A3 3(t) ] ) S3,, geneally satisfying the eesentations of Lemma 2 fo k = n = 3 and G ([ A3 3(t) ] ) S 3, +, in l2, -sheical coodinates with abitay (, ). Fo this uose, let R i (t) = x R3 : x = t t t λe (3) i, λ, i {, 2, 3}, denote the ays, which eesent the dashed edges of A3 3(t), see Figue 5(a). Note that, without loss of geneality, all figues ae dawn thoughout this oof fo = 3. Case : Let t. Because of (t, t, t)t / A3 3(t), the intesection [ A3 3(t) ] S 3, is emty if the oint (t, t, t)t is off o on the l 3, -unit shee, i.e. 3 t. Hence, [ A3 3(t) ] S 3, iff ( 3 t, ), and G ([ A3 3(t) ] ) S 3, + = fo evey R+, since A3 3(t) {x R 3 : x 3 }. The set G ([ A3 3(t) ] ) S3, is shown in Figue 6, whee P i = R i (t) S 3, and P i = G (P i ) fo i {, 2, 3}. Note that P = ( 2 t, t, t), P 2 = ( t, 2 t, t), and P 3 = ( t, t, ) 2 t. Figue 6: The set G ([ A3 3(t) ] ) ( ) S3, fo t and 3 t,. The ays stating in the oigin and assing though the oints (z, ) and ( z, t), and ( t, z) and (, z), esectively, enclose angles of the same magnitude α(ρ), whee one has to detemine z < such that the oint ( z, t) belongs to the l 2, -shee with -adius ρ, i.e. ρ = z + t. Thus, z = ρ t. By the definition of the tangent function, and making use of the notation at the beginning of Section 3., α(ρ) = α t, (ρ). 9

20 If ( 3 t, ), the set G ([ A3 3(t) ] ) S3, satisfies the eesentation ([ ] G A3 3(t) = SPH (2) = SPH (2) ) S3, ({(ρ, ϕ): ρ [ P 3, P ] (, ϕ π + α t, (ρ), 3π )}) 2 α t,(ρ) ({ [ 2 t (ρ, ϕ): ρ, ] }) t, ϕ (ϕ a, (ρ), ϕ b, (ρ)). Case 2: Let t >. We conside [ A3 3(t) ] S 3,. Case 2.: Let (, t). Then the l 3, -unit shee is comletely contained in A3 3(t). ) = K2,. Theefoe, G ([ A3 3(t) ] ([ S3,) = G A3 3(t) ] S 3, + Case 2.2: Let [t, 2t). This case occus iff the ays R i (t), i {, 2, 3}, do not intesect S 3,, but the thee lanes, which ae defined such that each of them contains exactly two of these ays, intesect the l 3, -unit shee, i.e. R i (t) S 3, =, and { ( λ)z + λz 2 R 3 : z R i (t), z 2 R j (t), λ [, ] } S 3, fo i, j {, 2, 3} with i j. In othe wods, the ange of fo this case ends if the ays ae tangents to S 3, and, without any loss of geneality, if R (t) is a tangent to S 3,, (, t, t) is the bounday oint. (a) G ([ A3 3(t) ] S 3, + ) (b) G ([ A3 3(t) ] S3, ) Figue 7: The sets in case 2.2. To achieve eesentations of the sets G ([ A3 3(t) ] ([ S 3,) + and G A3 3(t) ] ) S3, fo [t, 2t), see Figue 7. With the hel of l 2, -sheical coodinates, we detemine z < such that the oint ( z, t) belongs to the l 2, -shee with -adius ρ, i.e. ρ = z + ( t). Thus, z = ρ t. Analogously to the case t, the angle of the magnitude 2

21 α(ρ), enclosed by the ays stating in the oigin and assing though the oints (z, ) and ( z, t), satisfies α(ρ) = α t, (ρ). Since l 2, -shees ae invaiant with esect to otations of angles { k π : k 2 N} aound the oigin, the set G ([ A3 3(t) ] ) S 3, + satisfies fo t > and [t, 2t) the eesentation ([ ] ) ( ( ) ) G A3 3(t) S 3, + = K 2, \ K 2, t SPH (2) (M) whee M = { [ ] t (ρ, ϕ): ρ,, ϕ [α(ρ) π 2, π ] } 2 α(ρ) [α(ρ), π α(ρ)]. As the set A3 3(t) is unbounded with esect to the x 3 -diection, G ([ A3 3(t) ] ) S3, satisfies ([ ] ) G A3 3(t) S3, = K 2, \SPH (2) (M) fo [t, 2t). Using the symmety, one gets the claimed esult. Case 2.3: Let [ 2t, 3t). The ange of fo this case stats when the ays R i (t), i {, 2, 3}, ae tangents to S 3, and ends befoe the oigin (t, t, t) of the ays is on S 3,. Let {T i,, T i,2 } = R i (t) S 3, denote the set of oints of intesection of the ay R i (t) and the l 3, -unit shee and T i,j = G (T i,j ) fo i {, 2, 3} and j {, 2}. Analogously to the case t, T,j = ( ( ) j 2t, t, t), T 2,j = ( t, ( )j 2t, t), and T 3,j = ( t, t, ) ( )j 2t fo j {, 2}, and Ti, = T i,2, i {, 2, 3}, if = 2t. Figue 8 illustates that, as in the case befoe, the sets G ([ A3 3(t) ] ) S 3, + and G ([ A3 3(t) ] ) S 3, allow fo [ 2t, 3t) the eesentations ([ ] G A3 3(t) = SPH (2) SPH (2) ) S 3, + ({ ( (ρ, ϕ): ρ ({ (ρ, ϕ): ρ ] }) t,, ϕ (ϕ a,4 (ρ), ϕ b,4 (ρ)) [ t, ] }) 2t, ϕ (ϕ a,3 (ρ), ϕ b,3 (ρ)) and ([ ] G A3 3(t) = SPH (2) SPH (2) ) S3, ({ [ ] t (ρ, ϕ): ρ, ({ (ρ, ϕ): ρ }), ϕ (ϕ a,4 (ρ), ϕ b,4 (ρ)) ] [ t 2t,, ϕ (ϕ a,3 (ρ), ϕ b,3 (ρ)) }) K 2, ( t ). 2

22 (a) G ([ A3 3(t) ] S 3, + ) (b) G ([ A3 3(t) ] S3, ) Figue 8: The sets in case 2.3. Case 2.4: Let [ 3t, ), i.e. the ange of fo this case begins when the oint (t, t, t) is on the l 3,-unit shee. Unless fo = 3t, evey ay R i (t) has ecisely one oint of intesection with S 3, which is denoted by Q i, i {, 2, 3}. Then Q = ( 2t, t, t), Q 2 = ( t, 2t, t), and Q 3 = ( t, t, ) 2t. Let Q i = G (Q i ), i {, 2, 3}, and let the angle α(ρ) be defined as befoe. Figue 9 illustates that the sets G ([ A3 3(t) ] ([ S 3,) + and G A3 3(t) ] S3, as ) can be eesented (a) G ([ A3 3(t) ] S 3, + ) (b) G ([ A3 3(t) ] S3, ) Figue 9: The sets in case

23 ([ ] ) G A3 3(t) S 3, + ( = SPH ({(ρ, ϕ): ρ ] }) t,, ϕ (ϕ a,4 (ρ), ϕ b,4 (ρ)) and ([ ] ) G A3 3(t) S3, [ ] = SPH ({(ρ, ϕ): ρ t, [ SPH ({(ρ, ϕ): ρ t, }), ϕ (ϕ a,4 (ρ), ϕ b,4 (ρ)) ] }) ( ) 2t, ϕ (ϕ a,3 (ρ), ϕ b,3 (ρ)) K 2, t. 5.4 Maximum dfs as deivatives of aamete integals In this section, we establish the df of the maximum statistic in the case of thee deendent vs, see Coollay, taking the deivative of the aamete integal eesentation of the coesonding cdf given in Theoem. Poof of Coollay. Case : Let t <. Using the notation P(t, ) = 2 g()f(t, ), function f fom Theoem, and the Leibniz integal ule, f 3:3 (t) = 3 t 2 g() f t (t, ) d t 2 g( 3 t )f(t, 3 t ). Note that f(t, ( ) 3 t ) = and, because of α 2 t t, = π, 4 f (t, ) = t ( t) 2 t ρ ( ρ ) + 3 ( t ) 2 3π 2 αt,(ρ) dϕ t N 2 (ϕ) dρ π+α t,(ρ) 3π 2 βt, π+β t, dϕ N 2 (ϕ). Theefoe, in the case (t, ) (, ) ( 3 t, ), it follows that f (t, ) = k(t, ). t Case 2: Let t >. Let f, f 2, f 3, and f 4 denote the estictions of f (w..t. the vaiable ) to the intevals (, t), [t, 2t), [ 2t, 3t), and [ 3t, ), esectively, and 23

24 let P i (t, ) = 2 g()f i (t, ) and S i (t) = γ = α t, 2t (ρ) = actan ( 2ρ summand satisfies P i (t, ) d fo i =,..., 4. In this at of the oof, the fou summands S (t),..., S 4 (t) ae ( ) consideed seaately. Note that α t, 2t = π and lim α 4 t, (ρ) = π. We conside ρց t 2 ) and γ2 = α t, 3t (ρ) = actan ( 3ρ ). The fist and the second summand fulfills ds 2 dt (t) = 2t Futhe, P 2 (t, t) = ω 3, t 2 g(t), t ds dt (t) = ω 3,t 2 g(t), 2 g() f 2 t (t, ) d + 2 P 2 (t, 2t) P 2 (t, t). P 2 (t, 2t) = 2 2 t 2 g( ( ) 2t) ω 3, π γ π 2 ρ ( ρ ) (N (ϕ)) 2 dϕ dρ, and f 2 t (t, ) = ω t 3, 2 ( t ) 2 = ω 3, t ρ ( ρ t ( t ) 2 + 4k ρ,6 ( t, ). Taking the deivative of the fouth summand yields whee ds 4 dt (t) = ) 3t 2 g() f 4 t (t, ) d 3 P 4 (t, 3t) α t,(ρ)n 2 (π α t, (ρ)) dρ P 4 (t, 3t) = 3 2 t 2 g( 3t) ( ) 2 ω 3, π 2 +γ 2 ρ ( ρ ) (N π γ (ϕ)) 2 dϕ dρ π 2 +γ 2 ρ ( ρ ) (N π γ (ϕ)) 2 dϕ dρ γ 2 π 2 γ 2 ρ ( ρ ) (N (ϕ)) 2 dϕ dρ. 24

25 The atial deivative f 4 t satisfies the eesentation f 4 t (t, ) = ω 3, t 2 3 t ( t ) 2π() + k ρ,4 ( + k ρ,4 ( t, ) + k ρ,3( t, 2t ) + k ψ (π β t,, 3π 2 + β t,) t, ) whee k ρ and k ψ ae intoduced in Coollay. The thid summand S 3 satisfies ds 3 dt (t) = 3t 2t 2 g() f 3 t (t, ) d + 3 P 3 (t, 3t) 2 P 3 (t, 2t) whee P 3 (t, 2t) = P 2 (t, 2t) and P 3 (t, 3t) = P 4 (t, 3t). With f 3 (t, ) = 2t t α t,(ρ) π 2 αt,(ρ) it follows f 3 = f 4 + f 3 and f 3 t = f 4 t + f 3 t, whee ρ ( ρ ) (N (ϕ)) 2 dϕ dρ, f 3 t (t, ) = k ρ,3( t, 2t ) + k ψ( π 2 β t,, β t, ). Summaizing all intemediate esults, the coollay follows fom f 3:3 (t) = ds dt (t) + ds 2 dt (t) + ds 3 dt (t) + ds 4 dt (t). 5.5 Median fo n = 3 and Maximum fo n = 4 Following the same line as in the last two sections, we ove hee the eesentations of the cdf and the df of the median in the case of thee deendent vs, and the cdf of the maximum in the case of fou deendent vs. This oves Theoem 2, Coollay 2 and Theoem 3. Hee, calculations will not be given as detailed as in the eceding sections. Poof of Theoem 2. In an analogous manne as in the oof of Theoem, we use equation (4) fo the median statistic in the case of n = 3 and eesent the sets G ([ A3 2(t) ] ([ S3,) and G A3 2(t) ] ) S3,, given in Lemma 2 fo k = 2 and n = 3, fo an abitay (, ) using l 2, -sheical coodinates. In ode to do this, if t, 25

26 the cases to be distinguished ae (, 2 t ], ( 2 t, 3 t ], and ( 3 t, ). In the fist case, [ A3 2(t) ] S (+) 3, =. In the othe two cases, the gay-coloed sets shown in Figues have to be consideed. Hee, in contast to the oof of Theoem, but again without loss of geneality, figues ae dawn fo = 3 2. If t >, the diffeent cases ae (, 2t], ( 2t, 3t], and ( 3t, ). In the fist case, [ A3 2(t) ] S (+) 3, = S (+) 3,, and the sets G ([ A3 2(t) ] S (+) ) 3, in the othe cases ae shown in Figue. Note that thee is a helful symmety elation between the cases t and t >. Poof of Coollay 2. Taking the deivative of F 2:3 (t) diectly yields the df f 2:3 (t). Poof of Theoem 3. With the hel of equation (4), fo n = 4 and the maximum statistic, this oof follows analogously to that of Theoem o 2. The sets G ([ A4 4(t) ] ) S4, and G ([ A4 4(t) ] ) S4, given in Catesian coodinates by Lemma 2 need to be exessed using l 3, -sheical coodinates. To this end, we conside the seaate cases (, 4 t ] and ( 4 t, ), if t, and (, t), [t, 2t), [ 2t, 3t), [ 3t, 4t), and [ 4t, ), if t >. 6 Discussion In Mülle and Richte (25), the exact exteme value distibutions of the comonents of l 2, -symmetically distibuted andom vectos ae deived exlicitly. A efomulation in tems of skewed distibutions was oved in Batún-Cutz et al. (23). In the esent ae, assuming again the model class of continuous l n, -symmetic distibutions, the exact distibutions of ode statistics fo thee deendent and of exteme value statistics fo fou vs ae deived alying the geometic measue eesentation fom Richte (29) (a) ( 2 t, 3 t ] (b) ( 2 t, 3 t ] 26

27 diectly. In contast to othe alications of this geometic method, esults and oofs in the case of ode statistics become inceasingly involved if the dimension inceases. This exlains the need of finding a moe advanced method to make use of the geometic measue eesentation in highe dimensions. In the sheical case = 2, such a method was develoed fo n = 2 in Günzel et al. (22) and, genealizing this, fo abitay n in Richte and Venz (24). We hoe to eot a -genealization of this method in anothe ae. Refeences Aellano-Valle, R. B., & Genton, M. G. (27). On the exact distibution of linea combinations of ode statistics fom deendent andom vaiables. J. Multivaiate Anal., 98(), doi:.6/j.jmva Aellano-Valle, R. B., & Genton, M. G. (28). On the exact distibution of the maximum of absolutely continuous deendent andom vaiables. Stat. Pobab. Lett., 78, doi:.6/j.sl Aellano-Valle, R. B., & Richte, W.-D. (22). On skewed continuous l n, -symmetic distibutions. Chil. J. Stat., 3(2), Batún-Cutz, J., González-Faías, G., & Richte, W.-D. (23). Maximum distibutions fo l 2, -symmetic vectos ae skewed l, -symmetic distibutions. Stat. Pobab. Lett., 83(), doi:.6/j.sl Box, G. E., & Tiao, G. C. (973). Bayesian infeence in statistical analysis. Addison- Wesley Seies in Behavioal Science: Quantitative Methods. Reading, Mass. etc.: Addison-Wesley Publishing Comany. XVIII, 588. $ 9.6 (973). David, H. A., & Nagaaja, H. N. (23). Ode statistics (3d ed.). New Yok: Wiley. Dietich, T., Kalke, S., & Richte, W.-D. (23). Stochastic eesentations and a geometic aametization of the two-dimensional Gaussian law. Chil. J. Stat., 4(2), Fang, K.-T., Kotz, S., & Ng, K.-W. (99). Symmetic multivaiate and elated distibutions. London etc: Chaman and Hall. Gómez, E., Gómez-Villegas, M. A., & Maín, J. M. (998). A multivaiate genealization of the owe exonential family of distibutions. Commun. Stat., Theoy Methods, 27(3), doi:.8/ Günzel, T., Richte, W.-D., Scheutzow, S., Schicke, K., & Venz, J. (22). Geometic aoach to the skewed nomal distibution. J. Stat. Plann. Infeence, 42(2), doi:.6/j.jsi Guta, A. K., & Song, D. (997). l -nom sheical distibutions. J. Stat. Plann. Infeence, 6(2), doi:.6/s (96)

28 Jamalizadeh, A., & Balakishnan, N. (2). Distibutions of ode statistics and linea combinations of ode statistics fom an ellitical distibution as mixtues of unified skew-ellitical distibutions. J. Multiva. Anal., (6), doi:.6/j.jmva Kalke, S., & Richte, W.-D. (23). Simulation of the -genealized Gaussian distibution. J. Statist. Comut. Simulation, 83(4), doi:.8/ Kalke, S., Richte, W.-D., & Thaue, F. (23). Linea combinations, oducts and atios of simlicial o sheical vaiates. Comm. Stat., Theoy Methods, 42(3), doi:.8/ Moszyńska, M., & Richte, W.-D. (22). Revese tiangle inequality. Antinoms and semi-antinoms. Stud. Sci. Math. Hung., 49(), doi:.556/ssc- Math Mülle, K., & Richte, W.-D. (25). Exact exteme value, oduct, and atio distibutions unde non-standad assumtions. AStA Adv. stat. Anal., 99(), 3. doi:.7/s Nadaajah, S. (23). The Kotz-tye distibution with alications. Statistics, 37(4), doi:.8/ Nadaajah, S. (25). A genealized nomal distibution. J. Al. Stat., 32(7), doi:.8/ Nadaajah, S. (26). Acknowledgement of ioity: The genealized nomal distibution. J. Al. Stat., 33(9), doi:.8/ Osiewalski, J., & Steel, M. F. (993). Robust Bayesian infeence in l q -sheical models. Biometika, 8(2), Richte, W.-D. (27). Genealized sheical and simlicial coodinates. J. Math. Anal. Al., 336, doi:.6/j.jmaa Richte, W.-D. (28a). On l 2, -cicle numbes. Lith. Math. J., 48(2), doi:.7/s z Richte, W.-D. (28b). On the Pi-function fo nonconvex l 2, -cicle discs. Lith. Math. J., 48(3), doi:.7/s Richte, W.-D. (29). Continuous l n, -symmetic distibutions. Lith. Math. J., 49(), doi:.7/s Richte, W.-D. (24a). Classes of standad Gaussian andom vaiables and thei genealizations. In J. Knif & B. Pae (Eds.), Contibutions to mathematics, statistics, econometics, and finance ( ). Acta Wasaensia 296, Statistics 7. Richte, W.-D. (24b). Geometic disintegation and sta-shaed distibutions. J. Stat. Distib. Al. : 2. doi:.86/s Richte, W.-D. (25a). Convex and adially concave contoued distibutions. J. Pobab. 28

29 Stat.. (In evision.) Richte, W.-D. (25b). Nom contoued distibutions in R 2. In Lectue notes of Seminaio Intediscilinae di Matematica. Vol. XII. Univesity of Basilicata, Italy: Potenza: Seminaio Intediscilinae di Matematica (S.I.M.). (Acceted fo int.) Richte, W.-D., & Venz, J. (24). Geometic eesentations of multivaiate skewed ellitically contoued distibutions. Chil. J. Stat., 5(2), 7 9. Silveman, H. (994). The value of eoving. PRIMUS, 4(2), doi:.8/ Subbotin, M. (923). On the law of fequency of eo. Rec. Math. Moscou, 3(2),

30 (c) ( 3 t, ) (d) ( 3 t, ) Figue : The sets G ([ A3 2(t) ] ([ S 3,) +, on the left hand side, and G A3 2(t) ] ) S3,, on the ight hand side, if t. 3

31 (a) ( 2t, 3t] (b) ( 2t, 3t] (c) ( 3t, ) (d) ( 3t, ) Figue : The sets G ([ A3 2(t) ] ([ S 3,) +, on the left hand side, and G A3 2(t) ] ) S3,, on the ight hand side, if t >. 3