Exact distributions of order statistics of dependent random variables from ln,p-symmetric sample distributions, n ∈ {3,4}

Abstract Integral representations of the exact distributions of order statistics are derived in a geometric way when three or four random variables depend on each other as the components of continuous ln,psymmetrically distributed random vectors do, n ∈ {3,4}, p > 0. Once the representations are implemented in a computer program, it is easy to change the density generator of the ln,p-symmetric distribution with another one for newly evaluating the distribution of interest. For two groups of stock exchange index residuals, maximum distributions are compared under dependence and independence modeling.


Introduction
It is well known that uncorrelatedness of a nite number of random variables (rvs) implies their independence if their joint multidimensional distribution is a Gaussian one. More speci cally, if the density generating function (dgf) of a spherically distributed random vector is that of a Gaussian vector then the components of this vector are independent rvs. For any other choice of the dgf, these rvs depend on each other in a certain way. Similarly, if a random vector follows a continuous ln,p-symmetric or ln,p-spherical distribution, p > , its n components are independent if its dgf is that of a suitably de ned n-dimensional p-power exponential distribution, and only in this case.
Therefore, studying distributions of functions of spherically or ln,p-symmetrically distributed random vectors means, in general, studying distributions under speci c dependence assumptions w.r.t. the joint sampling distribution. Note that the class of l n, -symmetric distributions is just that of all spherical distributions. The type of dependence among the components of a continuous ln,p-symmetrically distributed random vector depends on both the dgf and the parameter p. One might call, for short, this dependence the ln,p-symmetry dependence.
Order statistics are useful tools in parametric and nonparametric statistics as well as, e.g., in reliability theory and other applied research areas. The distributions of order statistics of independent and identically distributed random variables are exhaustively dealt with in the last decades, see e.g. [8]. The probability density function (pdf) of the maximum statistic as well as that of a linear combination of order statistics of arbitrary absolutely continuous dependent random variables is studied in [2] and [1], respectively. In both papers, special emphasis is on the case that the joint multivariate sample distribution is an elliptically contoured distribution. In [14] and some papers referred to there, the latter investigations are followed up and further developed by representing the results with the help of skewed distributions. For a related result for continuous l ,p -symmetrically distributed sample vectors, see [5].
The class of elliptically contoured distributions extends that of spherical distributions, see [10]. Another extension is the class of ln,p-symmetric or ln,p-spherical distributions. This class has been introduced in [23], and dealt with later on, e.g., in [13]. A geometric measure representation of these distributions was proved in [27], see equation (3) in Section 6.1. This representation found applications to simulation in [16] and to the derivation of certain exact distributions in [17]. In [19], integral representations of exact distributions of extreme value statistics of l ,p -symmetrically distributed samples are proved making possible to easily change a given density generator with another one. Here, we extend these results to dimensions three and four.
These results and those presented here apply to di erent kinds of data analysis for data coming from various areas of statistical practice. If one is interested, e.g., in a long-term study of the behavior of the maximum distribution of the normalized residuals of a group of stock exchange indices then one may ask for a comparison of the observed distributions with those turning out from certain reasonably chosen models for the corresponding multivariate index residual distribution. The construction of well applicable multivariate probability distributions is one of the challenges of data analysis which is actively studied. A review of basic methods and results from such research is to be found, e.g., in [4], [7], [15], [21], [33], and [36]. Here, we follow the more recent construction of multivariate distributions in [27,28,31].
The further aim of the present paper, however, is twice. On the one hand, as indicated, we contribute new results on the exact distribution of order statistics of three or four dependent rvs if the sample distribution is an ln,p-symmetric one, n ∈ { , }. On the other hand, we conduce to a systematic study of cases in which the geometric measure representation successfully applies. While the rst aim of this paper needs no further explanation, the second one will to be discussed a bit closer in the following.
In [28], a problem is dealt with which was considered beforehand by several authors in a series of papers and by using di erent methods. Reproving certain of the known results with a new method, applying the geometric measure representation, actually needs most of the space in [28]. For the subsequent step of substantially extending the class of random variables possessing the same property of interest, however, only little additional e ort is needed. This way, a sometimes involved method suddenly mutates to a powerful ancillary tool of mathematical work. For a general discussion on the value of reproving, see [34].
Another e ect of systematically applying a geometric measure representation is discovered in [9]. Among other things, the authors develop a new integral representation of the cumulative distribution function (cdf) of the largest eigenvalue of a certain Wishart distributed random matrix although another representation in terms of hypergeometric functions has been well established already in the literature for a long time. The non-anticipated wage for these methodological e orts was in numerical stability properties of the new result. Furthermore, the systematic geometric measure theoretical studies in [12] and [32] bring more structure into a variety of well-known proofs and results on skewed distributions, and noticeably generalize several well established results.
In all these cases, new results are proved for rvs depending from each other under the in uence of a dgf and possibly additional parameters.
In order to summarize the two main aims of this paper, besides proving new results on distributions of order statistics, we extend the range where geometric measure representations successfully apply. This way, we contribute to establish such representations as standard ancillary tools of practical work in probability theory and statistics.
The rest of the present paper is organized as follows. As announced in this introductory part, in Section 2, real data of normalized residuals of certain stock exchange indices are analyzed for certain dependencies. In Section 3, general information on the model class of ln,p-symmetric distributions are given. Assuming this ln,p-symmetric model class, in Section 4, our main results on the cdf and pdf of maximum, median, and minimum statistics of three dependent rvs and on the cdf of extreme value statistics of four dependent rvs are presented. The pdf of the median is visualized, on the one hand, for trivariate p-generalized Gaussian distributed populations, p = , jointly with histogram plots for increasing sample sizes and, on the other hand, for l ,p -symmetrically Kotz type and Pearson Type VII distributed populations for several choices of parameters. Section 5 is aimed to discuss some gures given beforehand and to interrelate the underlying distributions with heavy tailed ones. In Section 6, the results of Section 4 are proved. In particular, basics of the geometric method of proof are explained in Section 6.1. In the nal Section 7, some conclusions are drawn from the results in the present paper.

Real data analysis and motivation
In this section, we basically analyze real data from international stock exchange indices for dependencies. In particular, we study normalized residuals of stock exchange indices DAX (Germany), Dow Jones Industrial Average (USA), and IBEX 35 (Spain) in Section 2.1 and of indices S&P TSE Composite Index (Canada), Hang Seng Index (Hong Kong), All Ordinaries Index (Australia), and RTS Index (Russia) in Section 2.2. The underlying data of the indices provided on quandl.com by Yahoo Finance is taken from 01-01-2000 until 12-31-2015 on all days (3262 in total) on which all stocks were opened and was downloaded on 01-26-2016.

. Analysis of three residuals
In Figure 1, the densities of the standardized stock exchange indices DAX, Dow Jones Industrial Average, and IBEX 35 are estimated using nonparametric kernel density estimation based upon the Epanechnikov kernel. In all of the gures, the solid red graphs are the estimations with the optimal bandwidth (bw) served by Matlab routine "ksdensity" and denoted henceforth by ϕ , ϕ and ϕ .
In Figure 2, estimated two-dimensional densities of pairs of the indices under consideration are presented together with their contour plots. These are done using a slight variation of Matlab function "Bivariate Kernel Density Estimation (V2.1)" ("gkde2.m" with File ID #19280) by Yi Cao downloaded from mathworks.com on 01-27-2016. To a certain extent, Figure 2 shows that the normalized residuals of DAX and IBEX 35 as well as of Dow Jones Industrial Average and IBEX 35 might be considered being not as strongly dependent as DAX and Dow Jones Industrial Average. Then, Figure 3(a) shows a comparison of the kernel density estimation of the maximum of normalized DAX and Dow Jones Industrial Average residuals (solid line) with the corresponding estimation when assuming a model of two independent random variables following the density ϕ(x) = ϕ (x) + ϕ (x) , x ∈ R, (dashed line). Note that the maximum pdf of two independent variables each having pdf ϕ and cdf Φ is        what will become more clear in the example considered in the next section. In Figure 3(d), a comparison of dependence and independence model of normalized residuals of all the three indices under consideration is shown. Again, the solid graph is the Epanechnikov kernel density estimation of the maximum data of normalized residuals of DAX, Dow Jones Industrial Average and IBEX 35 with optimal bandwidth served by "ksdensity". On the contrary, the dashed graph is the maximum pdf of three independent random variables having pdf ϕ(x) = ϕ (x) + ϕ (x) + ϕ (x) . In the three-dimensional independent case, this maximum pdf where Φ is the cdf regarding to ϕ. Because of Figure 3(d), the independence model seems to be not valid for this three-dimensional case at hand of maximum distributions. As a result, we have seen that it is not always easy to distinguish between independence and dependence models. In the next section, we are able to present, on the other hand, an example being close a "text-book-like" example for deciding this question.

. Analysis of four residuals
In analogy to Section 2.1, Figure 4 visualizes the estimated pdfs of normalized residuals of stock exchange indices S&P TSE Composite Index, Hang Seng Index, All Ordinaries Index, and RTS Index computed by nonparametric Epanechnikov kernel density estimation. First of all, the solid graphs corresponding to the estimation with optimal bandwidth and denoted by ψ , ψ , ψ , and ψ , respectively, are comparable in structure. Furthermore, in Figure 5, the estimated two-dimensional densities of all pairs of considered standardized stock exchange indices are illustrated together with their contours plots. These estimations are numerically done by bivariate kernel density estimation with Gaussian kernel and optimal choice of bandwidths using Matlab function "gkde2.m" by Yi Cao.     shows dependence among all pairs of indices. This impression is strongly supported by considering maximum distribution under independence and dependence modeling in Figure 6. The latter one compares the Epanechnikov kernel density estimation of the maximum statistic of normalized residuals of stock exchange indices under consideration (solid graph) with the exact maximum pdf x ∈ R, from four independent random variables whose pdf is estimated by Hence, the observations support the need of exact distribution of the maximum statistic and arbitrary order statistics, respectively, from dependent multivariate sample distributions.

The model class
We consider the model class of continuous ln,p-symmetric distributions in this paper as a subclass of the class of star-shaped distributions. This point of view leads to a slight change of notation for continuous ln,psymmetric distributions, compared with previous papers dealing with these distributions. Let K ⊂ R n be a star body having the origin in its interior and let S denote its topological boundary. The dr. According to [29], moreover assuming the homogeneity of degree one and a certain smoothness of h K , a probability measure having the pdf is called a star-shaped distribution with density contour de ning star body K, and denoted by Φ g,K . The normalizing constant allows the representation where O S denotes the star generalized surface measure on S and is de ned as well in [29]. If K is the unit ball of the nite-dimensional normed or antinormed space R n , · then h K (x) = x , and Φ g,K is called a norm or antinorm contoured distribution in R n , respectively, see [31] for the -dimensional and [30] for the general case. For the notion of an antinorm, we refer to [18].
Throughout this paper, let p > . We denote the ln,p-unit ball and the ln,p-unit sphere by Kn,p = {x ∈ stands for the p-functional. Then, h Kn,p (x) = |x|p, and h Kn,p is a norm if p ≥ and, according to [18], an antinorm if p ∈ ( , ). Further, the star generalized surface measure O Sn,p matches with the ln,p-generalized surface measure Op de ned in [27], and ωn,p denotes the ln,p-generalized surface content of Sn,p, ωn,p = In particular, an n-dimensional random vector X : Ω → R n de ned on a probability space (Ω, A, P) and having a pdf f X (x) = φ g,Kn,p (x) = g(|x|p) ωn,p I(g) , x ∈ R n , is said to follow the continuous ln,p-symmetric distribution Φg,p with dgf g. For short, the density f X = φ g,Kn,p is written as f X = φg,p. This pdf is norm contoured if p ≥ and radially concave star-shaped if p ∈ ( , ). From now on, we assume that g is speci cally chosen as a density generator (dg), i.e. the normalizing constant meets the condition ωn,p I(g) = . In other words, g is chosen in such a way that φg,p(x) = g(|x|p), x ∈ R n .
This notation of an ln,p-symmetric pdf slightly di ers from the notation f X (x) =g(|x| p p ), x ∈ R n , used in [13], [27], [3], [5], [17], [19], as well as [10] and [12] in the spherical case. Because of g(c) =g(c p ), c > , we obtain I(g) = I n,g,p where the notation I n,g,p = ∞ r n− g(r p ) dr is used in previous papers.
The remaining part of this section deals with examples of density generators of continuous ln,p-symmetric distributions. In slightly other notation, these and other examples can be found already in [13], and for the case p = in [10]. Note that only g = g PE in Example 3 generates independence of the components of the random vector. If p = , paying attention to the change of notation, this is the dg of standardized Kotz type distribution, see [20]. In [13], Φg Kt;M,β,γ ,p has parameter N = M − ≥ , and is called p-generalized Weibull distribution.
Example 3. The particular function g PE; = g PE is called the dg of the n-dimensional p-power exponential or p-generalized Gaussian or p-generalized Laplace distribution, If p = or p = , g PE is the dg of the n-dimensional Laplace or Gaussian distribution, respectively. Example 5. The dg g St;ν of the ln,p-symmetric Student-t distribution with ν > degrees of freedom is de ned as In addition, g St;ν = g PT ; n+ν p ,ν . Example 6. The dg g C of the ln,p-symmetric Cauchy distribution satis es as it is well known in the spherical case p = .
otherwise denote the indicator function of the set A.
Example 7. The dg of the ln,p-symmetric Pearson Type II distribution with parameter ν > is

Exact distributions of order statistics
For the rest of the paper, we assume that the random variables X , . . . , Xn are the components of the random vector X, X ∼ Φg,p, for an arbitrary shape/ tail parameter p > as well as an arbitrary dg g. Furthermore, we denote the corresponding vector of order statistics by X (ord) (n) = (X :n , . . . , Xn:n) T and the cdf and pdf of X k:n , k = , . . . , n, by F k:n and f k:n , respectively. The following result describes the basic structure of our representations of F k:n and f k:n .
Lemma 1 (Separating property). The cdf and the pdf of X k:n allow the representations t ∈ R, with functions f , h : R × ( , ∞) → R not depending on the dg g.
For simplicity of notation, we do not indicate that the functions f and h depend on the integers n and k, and on the parameter p. Note that the in uence of g on the distribution of X k:n is separated in Lemma 1 from that of all the other parameters. Once the functions f and h are implemented in a computer program, it is easy to change a certain dg g with another one for newly evaluating the functions F k:n and f k:n . Figures 7-9 show the median density for di erent types of the dg (recognize di erent scaling in di erent pictures). The underlying results of Sections 4.1 and 4.2 specify the functions f and h, and will be derived on using the geometric measure representation (3), see Section 6.1.

Corollary 1. The pdf of the maximum statistic satis es representation (2) with n = k = and
) .
It is worthwhile to mention that the cdf and the pdf of the minimum statistic satisfy the representations F : (t) = − F : (−t) and f : (t) = f : (−t), t ∈ R, respectively.

Theorem 2. The cdf of the median statistic satis es representation (1) with n = , k = and
Corollary 2. The pdf of the median statistic satis es representation (2) with n = , k = and  (2) and Corollary 2). Note that the numerical correctness of our evaluations is revealed by adding histogram plots of samples of increasing sizes from up to . × . Also illustrating the pdf of the median statistic but now of three dependent rvs following a joint continuous l ,p -symmetric distribution, Figures 8 and 9 deal with general dependence generating dgs of Kotz type and Pearson Type VII, respectively. Note the di erent scales of axes of ordinates as well as of abscissas, and that Figures 9(a) and 9(b) are further discussed under several aspects in Section 5.

Heavy tails
Distributions having heavy tails play an important role in statistical practice and especially nd many applications to insurance and nancial mathematics. The median pdf f : plotted in Figure 9 deals with heavy tails where the dg of X is of l ,p -symmetric Pearson Type VII which includes both Student and Cauchy type sample distributions. It appears to be typical in such cases that only very few probability mass is concentrated around the distribution center leading on the right hand sides of Figures 9(a) and 9(b) to the misleading impression that the drawn densities could build a monotonically decreasing sequence of functions. By zooming into the right hand side of Figures 9(a), however, and taking the symmetry w.r.t. axis of ordinates into account, one detects the points of intersection of the black and the green solid, the black and the green dashed, and the green solid and the green dashed graphs at t / ≈ ± , t / ≈ ± , and t / ≈ ± , respectively, see Figure  10. A similar explanation avoids a potential misunderstanding in the case of Figure 9(b). Furthermore, the Figures 9(a) and 10 suggest the visual impression that the tail heaviness of the distribution of the median statistic of three dependent rvs following a joint l , -symmetric Pearson Type VII distribution increases if the parameter M is constant and the parameter ν > increases. For speci c values of interval probabilities see Table 2 where the values Aν(z) = If M is constant and ν > increases, such a manner of heaviness of tails can be observed in all cases of Figure 9.

Proofs
In order to proof the assertions from Section 4, the general method of proof and some basics on applying this method to order statistics are presented in Sections 6.1 and 6.2, respectively. Afterwards, the claimed results on the distributions of order statistics for three dependent rvs and extreme value statistics for four dependent rvs are established whereas the details of proofs decrease in quantity in later parts of these sections.

. Basics of the geometric method of proof
Let T : R n → R be any statistic and A(t) = x ∈ R n : T(x) < t a sublevel set (sls) generated by it. If X ∼ Φg,p with an arbitrary dg g, the cdf of T(X) is The geometric measure representation in [27], with notations as described in Section 3 suitably adapted to the ones used in [29] and here, applies According to [27], the ln,p-generalized surface content is de ned on B n by : [ , ∞) × [ , π) ×(n− ) × [ , π) → R n− and its corresponding inverse mapping, see [24], the cdf of T(X) allows the general representation

. General representations of the domains of integration M +(−) (t,r)
As it can be seen from Section 6.1, studying the sets G( r A(t) ∩ S − n,p ) and G( r A(t) ∩ S + n,p ) plays a fundamental role for the application of the geometric measure representation formula. The present section is aimed to prove general representations of these sets if the generating statistic is any order statistic. More speci c representations will be derived from it in the next section and will be used there to prove results for all cases considered in Sections 4.1 and 4.2.
For k ∈ { , . . . , n}, let A n k (t) = x ∈ R n : at least k components of x are less than t , t ∈ R, be a sls generated by the kth order statistic of an n-dimensional random vector. An illustration of the set A n k (t) can be seen in Figure 11 for (n, k) ∈ {( , ), ( , )} and t < .
Remark 1. Let X be a continuous and symmetrically with respect to the origin distributed random vector, X ∼ −X, and let F k:n (t) = P (X k:n < t) be the cdf of the kth order statistic X k:n of X. Then, for k = , . . . , n and every t ∈ R, F n−k+ ,n (t) = − F k,n (−t), and F k:n (t) = P X ∈ A n k (t) .

Lemma 2. If t ≤ , then
and, if t > , x y z (a) Sls of the maximum where A m l (t) = ∅ if l = or l > m,Å denotes the topological interior of the set A ⊆ R n , Kn,p(ρ) = {x ∈ R n : |x|p ≤ ρ} the ln,p-ball with p-radius ρ ∈ ( , ∞), and the ln,p-layer with p-radii ρa < ρ b .
In the following Sections 6.3 as well as 6.5, Lemma 2 is particularly utilized for n ∈ { , }. Despite that Lemma 2 and its proof, see Appendix A, are given for an arbitrary integer n since their formulations are identical.
The next step of analyzing the sets G( r A(t) ∩ S − n,p ) and G( r A(t) ∩ S + n,p ) consists of numerous case studies. Because the number of cases increases if the number of rvs does, we restrict the outline of this way mainly to the case of three rvs.

. Speci c representations of the domains of integration for considering the maximum of three dependent rvs
This section demonstrates that, in the case of three dependent rvs, the geometric method of proof applies as successful as in [19] where the case of sample size two was dealt with. The present calculations may also serve as an orientation for the derivation of analogous results in more general star-shaped model classes.
Proof of Theorem 1. To get the exact cdf of the considered statistic, according to equation (4), it remains to represent the sets G r A (t) ∩ S − ,p , generally satisfying the representations of Lemma 2 for k = n = and G r A (t) ∩ S + ,p , in l ,p -spherical coordinates with arbitrary r ∈ ( , ∞). For this purpose, let i ∈ { , , }, denote the rays, which represent the dashed edges of r A (t), see Figure 11(a). Note that, without loss of generality, all gures are drawn throughout this proof for p = .
Case 1: Let t ≤ . Because of r (t, t, t) T ∉ r A (t), the intersection r A (t) ∩ S ,p is empty if the point r (t, t, t) T is o or on the l ,p -unit sphere, i.e. ≤ r p √ |t|. Hence, r A (t) ∩ S ,p ≠ ∅ i r ∈ p √ |t|, ∞ , and G r A (t) ∩ S + ,p = ∅ for every r ∈ R+, since r A (t) ⊂ x ∈ R : x ≤ . The set G r A (t) ∩ S − ,p is shown in Figure 12, where P i = R i (t) ∩ S ,p and P i = G (P i ) for i ∈ { , , }. Note that P = − r p r p − |t| p , r t, r t , P = r t, − r p r p − |t| p , r t , and P = r t, r t, − r p r p − |t| p . Figure 12: The rays starting in the origin and passing through the points (z, ) and z, r t , and r t, z and ( , z), respectively, enclose angles of the same magnitude α(ρ), where one has to determine z < such that the point z, r t belongs to the l ,p -sphere with p-radius ρ, i.e. ρ p = |z| p + r t p . Thus, z = − r p ρ p r p − |t| p . By the de nition of the tangent function, and making use of the notation at the beginning of Section 4.1, α(ρ) = α t,r (ρ).
If r ∈ p √ |t|, ∞ , the set G r A (t) ∩ S − ,p satis es the representation Case 2: Let t > . We consider r A (t) ∩ S ,p . Case 2.1: Let r ∈ ( , t). Then the l ,p -unit sphere is completely contained in r A (t). Therefore, . This case occurs i the rays R i (t), i ∈ { , , }, do not intersect S ,p , but the three planes, which are de ned such that each of them contains exactly two of these rays, intersect the l ,p -unit sphere, i.e. R i (t) ∩ S ,p = ∅, and In other words, the range of r for this case ends if the rays are tangents to S ,p and, without any loss of generality, if R (t) is a tangent to S ,p , r ( , t, t) is the boundary point. To achieve representations of the sets G r A (t) ∩ S + ,p and . Median for n = and Maximum for n = Following the same line as in the last two sections, we prove here the representations of the cdf and the pdf of the median in the case of three dependent rvs, and the cdf of the maximum in the case of four dependent rvs. This proves Theorem 2, Corollary 2 and Theorem 3. Here, calculations will not be given as detailed as in the preceding sections.
Proof of Theorem 2. In an analogous manner as in the proof of Theorem 1, we use equation (4) for the median statistic in the case of n = and represent the sets G r A (t) ∩ S − ,p and G r A (t) ∩ S − ,p , given in Lemma 2 for k = and n = , for an arbitrary r ∈ ( , ∞) using l ,p -spherical coordinates. In order to do this, if t ≤ , the cases to be distinguished are r ∈ ( , p √ |t|], r ∈ ( p √ |t|, p √ |t|], and r ∈ ( p √ |t|, ∞). In the rst case, r A (t) ∩ S −(+) ,p = ∅. In the other two cases, the grey-colored sets shown in Figures 16 have to be considered. Here, in contrast to the proof of Theorem 1, but again without loss of generality, gures are drawn for p = .
If t > , the di erent cases are r ∈ ( , p ,p in the other cases are shown in Figure 17. Note that there is a helpful symmetry relation between the cases t ≤ and t > . Proof of Corollary 2. Taking the derivative of F : (t) directly yields the pdf f : (t).
Proof of Theorem 3. With the help of equation (4), for n = and the maximum statistic, this proof follows analogously to that of Theorem 1 or 2. The sets G r A (t) ∩ S − ,p and G r A (t) ∩ S − ,p given in Cartesian coordinates by Lemma 2 need to be expressed using l ,p -spherical coordinates. To this end, we consider the separate cases r ∈ ( , p √ |t|] and r ∈ ( p √ |t|, ∞), if t ≤ , and r ∈ ( , t), r

Discussion
In [19], the exact extreme value distributions of the components of l ,p -symmetrically distributed random vectors are derived explicitly. A reformulation in terms of skewed distributions was proved in [5]. In the present paper, assuming again the model class of continuous ln,p-symmetric distributions, the exact distributions of order statistics for three dependent and of extreme value statistics for four rvs are derived applying the geometric measure representation from [27] directly. In contrast to other applications of this geometric method, results and proofs in the case of order statistics become increasingly involved if the dimension increases.
(a) r ∈ ( p √ t, This explains the need of nding a more advanced method to make use of the geometric measure representation in higher dimensions. In the spherical case p = , such a method was developed for n = in [12] and, generalizing this, for arbitrary n in [32]. We are going to report a p-generalization of this method in another paper.
Another possible further step could be to extend the model class to p-generalized elliptically contoured distributions, see [29], and to study exact distributions of order statistics from these more general multivariate distributions.
Summarizing the results given above, the rst assertion of the lemma follows. The other cases can be dealt with in an analogous way.