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On (p_{1},…,p_{k})spherical distributions
 WolfDieter Richter^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s404880190097z
© The Author(s) 2019
 Received: 14 January 2019
 Accepted: 23 May 2019
 Published: 12 June 2019
Abstract
The class of (p_{1},…,p_{k})spherical probability laws and a method of simulating random vectors following such distributions are introduced using a new stochastic vector representation. A dynamic geometric disintegration method and a corresponding geometric measure representation are used for generalizing the classical χ^{2}, t and Fdistributions. Comparing the principles of specialization and marginalization gives rise to an alternative method of dependence modeling.
Keywords
 (p _{1},...,p _{k})power exponential distribution
 simulation of (p _{1},...,p _{k})spherical uniform distribution
 signinvariant distribution
 (p _{1},...,p _{k})spherical surface content measure
 (p _{1},...,p _{k})spherical radius variable
 matrix times vector stochastic representation
 dynamic disintegration
 (p _{1},...,p _{k})spherical coordinates
 generalized χ ^{2}
 t and Fdistributions
 dependence modeling
 specialization vs. marginalization
 specialization Copula density
 poly Beta function
 angular Beta density
Mathematics Subject Classification (2000)
 60E05
 62E15
 60D05
 14J29
 28A50
 28A75
Introduction
A basic notion from the theory of spherical probability laws is that of the stochastic basis which is a random vector following the uniform distribution on the Euclidean unit sphere in the kdimensional Euclidean space \(\mathbb {R}^{k}\), see e.g. Fang et al. (1990). In this monograph, multivariate uniform distributions are introduced in an algebraic way without referring to any type of surface measure. Numerous authors deal with the (singular) uniform distribution by considering the density of its k−1dimensional marginal distribution, an approach which will, however, not further be discussed, here. Instead, the point of view of uniformity of which is the speech here is to define it by having a constant RadonNikodym derivative with respect to the Euclidean surface content measure. This geometric view onto the class of spherical distributions is the background of the corresponding geometric measure representation (2) in Richter (1991). This representation extends the one given in (3) in Richter (1985) for the Gaussian law and was exploited later on in a series of papers on probabilities of large deviations and on various statistical distributions. For a related survey see e.g. Richter (2015b).
Many authors studied more general or modified multivariate distribution classes. Just for getting an impression of this research area taking into account different points of view we refer to the more recent papers Field and Genton (2006), Arnold et al. (2008), Kamiya et al. (2008), Balkema et al. (2010), Balkema and Nolde (2010), Richter (2014), Richter (2015a) and Nolan (2016) as well as the references given there.
A detailed geometric description of uniformity of the stochastic basis of l_{k,q}spherical distributions, q>0,k∈{2,3,…}, is given in Richter (2009). The mathematical background there is based upon both a study of the different notions of Euclidean and l_{k,q}surface content measures on l_{k,q}spheres and on using suitable coordinates for evaluating the uniform measure of given subsets of spheres. The coordinates mentioned were introduced in Richter (2007) solving a long standing problem apparently conclusively treated as insolvable in Szablowski (1998).
here. As a consequence, distributions studied here are not invariant w.r.t. the class of orthogonal transformations or at least w.r.t. particular rotations but still appear to be signinvariant meaning invariance w.r.t. multiplication with sign matrixes. While sample schemes for identically distributed variables automatically imply exchangeability of all variables in the present paper this property is excluded due to assumption (1). Distributions considered here are therefore fully outside the scope of the spherical distribution part and in consequence of the remaining parts of the classical monograph by Fang et al. (1990) and numerous work following it.
The present paper is structured as follows. The class of pspherical uniform distributions is introduced in a geometrically motivated way in Section 2 and extended to the class of pspherical or l_{k,p}symmetric distributions in Section 3.1. The Sections 3.2 and 3.3 deal with a geometric measure representation and a combination of the principles of specialization and marginalization in dependence modeling, respectively.
The class of pspherical uniform distributions
Definition 1
Here, the qgeneralized trigonometric functions \( \sin _{q}\varphi =\frac {\sin \varphi }{N_{q}(\varphi)}\) and \( \cos _{q}\varphi =\frac {\cos \varphi }{N_{q}(\varphi)} \) with N_{q}(φ)=( sinφ^{q}+ cosφ^{q})^{1/q}, q>0 are introduced in Richter (2007) and used in studying and geometrically representing generalized spherical power exponential distributions in a series of papers starting from Richter (2009).
Remark 1
Remark 2
The following definition is well motivated by the fact that for the case p=(q,…,q) (not considered here) the l_{k,q}surface content measure is similarly introduced in Richter (2009) and proved to be equivalent to the differentialgeometric definition. Much more general results of this type of equivalence are proved for norm and antinorm spheres in Richter (2015a).
Definition 2
Definition 3
Remark 3
The notion of S_{p}surface content of D_{p}(r)A is different from the notion of Euclidean surface content of D_{p}(r)A (unless for the case p=(2,...,2) which, however, because of assumption ( 1 ) is not allowed to appear, here).
Definition 4
This definition corresponds to Definition 8 in Richter (2007) where, however, p is a scalar while it is a kdimensional vector, here. Similarly, the notation \(U_{p}=D_{p}\left (\frac {1}{R^{(p)}}\right)X\) introduced above is closely connected to that given in the same paper where p is a scalar. However, while \(R^{(p)}=\frac {X_{1}^{p_{1}}}{p_{1}}+...+\frac {X_{k}^{p_{k}}}{p_{k}}\) denotes a certain ”mixedpowerofradius”, here, not being the power of a norm or antinorm, in Richter (2007) the symbol R actually means a norm or antinorm. The proof of the following theorem is analogous to that of Theorem 2 in Richter (2017) and will therefore be omitted, here.
Theorem 1

(a) The random vector U_{p} follows the pspherical uniform distribution, U_{p}∼ω_{p}, is independent of R^{(p)}, and R^{(p)} has the following density with respect to the Lebesgue measure on the real line$$ \left(\frac{1}{p_{1}}+...+\frac{1}{p_{k}}\right)r^{\frac{1}{p_{1}}+...+\frac{1}{p_{k}}1}I_{[0,1)}(r) dr. $$(12)

(b) If, vice versa, ξ and W are independent where ξ has density ( 12 ) and W∼ω_{p} then η=D_{p}(ξ)·W is uniformly distributed on the unit ball B_{p}.
We are now in a position to disclose the basic message of this paper as follows: matrixmultiplication of a pspherical uniformly distributed random vector U_{p} by D_{p}(R) where U_{p} and the random variable R≥0 are independent generates the world of distributions being of main interest, here.
Remark 4
If the random variable Y is uniformly distributed on (0,1) then \(Z_{p}=Y^{1/\left (\frac {1}{p_{1}}+...+\frac {1}{p_{k}}\right)}\) follows the density ( 12 ), that is \(Z_{p}\overset {d}{=}{R^{(p)}}.\) Thus R^{(p)} can be simulated by Z_{p}.
Theorem 2
Let the random vector X be uniformly distributed on the unit pball B_{p}. Then the pspherical radius variable R^{(p)} and the pspherical angles Φ_{1},...,Φ _{k−1} of X are independent and the angle Φ_{i} follows the angular Beta density with parameters \({\mathfrak {p}}, a_{i}\) and \(b_{i}, f_{{\mathfrak {p}},a_{i},b_{i}}\), where \(a_{i}=\frac {\mathfrak {p}}{p_{i}}, b_{i}=\frac {\mathfrak {p}}{p_{k}}+...+\frac {\mathfrak {p}}{p_{i+1}}, i=1,...,k2\) and \(a_{k1}=\frac {\mathfrak {p}}{p_{k1}}, b_{k1}=\frac {\mathfrak {p}}{p_{k}} \).
Proof
Now, Definition 3 applies. □
Remark 5
Simulation Algorithm 1[pspherical uniform distribution inS_{p},p=(p_{1},...,p_{k})^{T}] Step 1 For i∈{1,…,k1}, simulate (V_{i,1},V_{i,2})∼U([0,1]×[0,1])until \(V_{i,1}^{p_{i}}+V_{i,2}^{1/\left (\frac {1}{p_{i+1}}+\ldots +\frac {1}{p_{k}}\right)}\leq 1\). Step 2 Calculate \(W_{i}=\frac {V_{i,1}^{p_{i}}}{V_{i,1}^{p_{i}}+V_{i,2}^{1/\left (\frac {1}{p_{i+1}}+\ldots +\frac {1}{p_{k}}\right)}}, i=1,...,k1\). Step 3 Simulate independently (S_{1},…,S_{k})∼U({−1,+1}^{×k}). Step 4 Calculate \(U_{p,i}=S_{i}\left (\prod \limits _{j=1}^{i1}(1W_{j})W_{i}\right)^{1/p_{i}}\) for i=1,…,k−1
and \(U_{p,k}=S_{k}\prod \limits _{j=1}^{k}(1W_{j})^{1/p_{k}}\). Step 5 Return U_{p}=(U_{p,1},...,U_{p,k})^{T}.
Simulation Algorithm 2 [Uniform distribution inB_{p},p=(p_{1},...,p_{k})^{T}] Step 1 Simulate U_{p}∼U(S_{p}) according to Algorithm 1. Step 2 Simulate independently: Y∼U(0,1).
Calculate \({R^{(p)}}=Y^{1/(\frac {1}{p_{1}}+\ldots +\frac {1}{p_{k}})}\). Step 3 Return X=R^{(p)}·U_{p}
Remark 6
Remark 7

(a) Let us callthe D_{p}transformedsector measure on \(\mathfrak B(S_{p})\) (or, more precisely, the uniform probability measure of the D_{p}transformed sector Se(A,1) of B_{p}). It follows from the obvious equations$$sm_{p}(A)=\frac{\mu(Se_{p}(A,1))}{\mu(B_{p})} $$that$$\frac{O_{P}(A)}{O_{P}(S_{p})}=\frac{f_{A}'(1)}{f_{S_{p}}'(1)} =\frac{\pi_{p}^{*}(A)}{\pi_{p}^{*}(S_{p})}=\frac{\mu(Se_{p}(A,1))}{\mu(Se_{p}(S_{p},1))} =\frac{\mu(Se_{p}(A,1))}{\mu(B_{p})} $$Thus, the pspherical uniform probability law ω_{p} can also be called a D_{p}transformedsector measure. For an interpretation of ω_{p} as cone measure see (e) and Remark 9 (b).$$\omega_{p}(A)=sm_{p}(A). $$

(b) According to Remark 3, the notion of pspherical uniform distribution on \(\mathfrak B(S_{p})\) is different from the notion of uniform distribution with respect to Euclidean surface content (unless for p_{1}=...=p_{k}∈{1,2,∞}).

(c) Fine properties of the Euclidean surface content measure defined on the Borel σfield of the Euclidean unit sphere, a precursor of the S_{p}surface content measure O_{p} considered here, are exploited by the author in the eighties and nineties of the last century in a series of papers on large deviations. A main idea in the background of those work is the development and application of a generalized method of indivisibles extending a classical approach by Cavalieri and Torricelli, see Richter (1985), Richter (2015b) and Günzel et al. (2012).

(d) It is a challenging problem to find a differentialgeometric interpretation of O_{p} as it was found in Richter (2009) for the l_{k,q}surface content measure and in Richter (2015a) for norm and antinorm spheres. This problem was first stated for the twodimensional case in Richter (2017).

(e) Several authors who study uniform distributions on (non D_{p}transformed) generalized spheres make use of the notion cone measure instead of sector measure and rely on the last representation in (a), that is on relating volumes to each other, see e.g. Naor and Romik (2003) and Barte et al. (2005) and a series of follow up papers.

(f) Only a few days before finishing the present paper, Amir AhmadiJavid kindly let me know his joint article AhmadiJavid and Moeini (2019) where the authors follow another way of considering a uniform distribution on \(\mathfrak B(S_{p})\). They start, in Definition 2.3, with a random vector being uniformly distributed in a parallelepiped and, referring to the work of Schechtman and Zinn (1990), Rachev and Rüschendorf (1991), Song and Gupta (1997), Liang and Ng (2008), Harman and Lacko (2010) and Lacko and Harman (2012), later make use of a common (nondynamical) notion of cone measure (just like the one mentioned in (e)) for studying a certain type of uniform distributions on \(\mathfrak B(S_{p})\). A closer comparison with the method presented here, where we start with a uniform distribution on a ball B_{p} and continue with a dynamically transformedcone measure, would be of interest for future work.
The class of pspherical distributions
A random vector distributed according to the pspherical uniform distribution builds the stochastic basis of any pspherical distributed random vector considered in Section 3.1. Examples of light and heavy distribution centers and tails are possible. A geometric measure representation and its applications allow studying exact distributions of generalized χ^{2}, t and Fstatistics in Section 3.2. The final Section 3.3 gives a sketch of an alternative approach to describing dependence of random variables following onedimensional specializations of kdimensional distributions instead of marginal distributions.
3.1 Definitions and Examples
Definition 5
is said to follow the pspherical distribution \(\Phi _{p}^{cdf(F)}=\Phi _{p}^{cf(\phi)}\). The vector U_{p} is called the pspherical uniform basis and R the generating variate of X, and (15) a stochastic representation of X. The distribution of X will alternatively be denoted \(\Phi _{p}^{pdf(f)}\) if R has probability density function (pdf) f.
Remark 8
If \(\mathbb {E} \left (R^{(p)\frac {1}{\min \{p_{1},...,p_{k}\}}}\right)\) is finite then, due to Remark 6, \(\mathbb {E}(X)=0_{k}\) and if \(\mathbb {E} \left (R^{(p)\frac {2}{\min \{p_{1},...,p_{k}\}}}\right)<\infty \) then \(\mathbb V(X_{i})=\mathbb {E}\left (R^{(p)\frac {2}{p_{i}}}\right)\mathbb V (U_{p,i}), i=1,...,k.\) For the derivation of moments in the case p_{1}=...=p_{k} we refer to ArellanoValle and Richter (2012).
Theorem 3
Proof
Corollary 1

(a) The distribution of a pspherically distributed random vector X is uniquely determined by the distribution of its generating variate R.

(b) If a pspherically distributed random vector X has a density, then it is of the form f_{X}=φ_{g;p},where g:[0,∞)→[0,∞) is a density generating function (dgf) satisfying$$ \varphi_{g;p}(x)= C(g;p)g\left(x^{(p)}\right), x\in\mathbb{R}^{n} $$and the normalizing constant allows the factorization$$ 0<I(g;p)=\int\limits_{0}^{\infty} r^{\frac{1}{p_{1}}+...+\frac{1}{p_{k}}1}g(r)dr<\infty, $$This density is invariant w.r.t. multiplication with sign matrices, or signinvariant or signsymmetric. For a general class of symmetric distributions we refer to ArellanoValle et al. (2002) and ArellanoValle and del Pino (2004).$$ \frac{1}{C(g;p)}=I(g;p)O_{p}(S_{p}). $$
Both this result and the following definition transfer earlier statements from Richter (2014) to the present case. The following definition adopts notation in Müller and Richter (2016) and is aimed to make the notion of dgf unique.
Definition 6
is called density generator (dg) of a continuous pspherical distribution.
Example 1
Example 2
3.2 Geometric measure representation
The following theorem is a geometricmeasure theoretic counterpart to Theorem 3. Its proof follows the line of author’s earlier work in Richter(1985, 1991, 2014, 2017). The main aim of this section is to present first applications of the geometric measure representation extending the classical HelmertPearson χ^{2}, Gosset alias Student t and Fisher Fdistributions.
Let \(\Phi _{g}^{(p)}\) denote the continuous pspherical distribution law having dg g.
Theorem 4
Example 3
Differently from the χ^{2}(k)distribution where the parameter k corresponds to the dimension of a subspace of the sample space, here, the parameter \(\frac {1}{p_{1}}+...+\frac {1}{p_{k}}\) itself does not allow interpretation as dimension of a linear space or a linear subspace of the sample space, but its number of summands k does.
Note that the density of T was dealt with for the particular dg of the generalized Gaussian law in Taguchi (1978). For \(\frac {1}{p_{1}}+...+\frac {1}{p_{k}}=\frac {k}{2}\) the pdf in ( 20 ) is called the ggeneralization of the Chisquare density with k degrees of freedom (d.f.) in Richter (1991) (note that there holds p_{1}=…=p_{k} but ( 16 ) is not assumed to be satisfied there). For more partial cases and statistical applications of this distribution, see Richter (2007, 2009, 2016).
Similarly, if we put \(g=g^{(p)}_{Kt;M,\beta,\gamma }\) or \(g=g^{(p)}_{PT7;M,\nu }\) in ( 19 ) then \(\phantom {\dot {i}\!}f_{R^{(p)}}\) and \(\mathbb V(X_{i})\) will be correspondingly specified.
Example 4
We call the pdf in ( 23 ) the Fisherp density with (m,k−m) d.f. or F_{m,k−m}(p)density, for short.
Example 5
Note that, as in Example 4, B_{T} is a D_{p}transformedcone or curvedtransformedcone type set satisfying ( 22 ).
Remark 9

(a) Because Fisherp and Studentp distributions do not depend on the dg, the Fp and tp statistics are called grobust. For a study of gsensitivity and grobustness of certain statistics see Ittrich et al. (2000) and for a study of statistics generating curvedtransformedcone type sets, see Ittrich and Richter (2005).

(b) Let B_{T}(λ) be a curvedtransformedcone type set satisfying ( 22 ) and put A=B_{T}(λ)∩S_{p}. By Theorem 4, \(\Phi ^{(p)}_{g}(B_{T}(\lambda))=\omega _{p}(A).\) It is reasonable therefore to call C^{(p)}(A)=ω_{p}(A) the D_{p}transformedcone measure of \(A\in \mathfrak B(S_{p})\).

(c) We recall that several representations of Student distributed statistics were given in Richter (1995). In particular, the two facts are exploited there that the ipf of the cone {T<λ} does not depend on its radius variable and that the multivariate standard Gaussian law is invariant w.r.t. orthogonal transformations, together leading to grobustness of Fisher’s and Student’s statistics. Due to assumption ( 22 ), we observe in the present situation that the ipf of B_{T}(t) does not depend on the generalized radius variable and we observe invariance of Fisher’s and Student’s statistics w.r.t. any transformation D_{p}(r) where r>0 is used to prove grobustness of T.Thus, if an arbitrary statistic T generates sublevel sets satisfying assumption ( 22 ) then such statistic is grobust.
3.3 Dependence modeling: specialization vs. marginalization
Imagine now each of k experimenters observe another random variable, let them combine their (possibly dependent) observations by a vector and describe this vector by a joint cdf F^{(k)} (possibly including dependence). In hindsight, did the experimenters construct the multivariate cdf F^{(k)} starting from the specializations F_{1},...,F_{k} of F^{(k)} or from the marginal cdfs \(F_{1}^{*},...,F_{k}^{*}\) of F^{(k)}? In other words, are the experimenters searching for a multivariate cdf F^{(k)} such that their original observations follow marginal distributions or specializations of F^{(k)}? For an illustration, in the case p_{1}=...=p_{k} and at hand of certain stock exchange indices, see Müller and Richter (2016).
The following definition is therefore well motivated.
Definition 7
We call c_{sp} the specialization copula density.
Example 6
Example 7
Remark 10
The univariate qgeneralized normal distribution or qpower exponential distribution has been parameterized in different ways in the literature, for a recent survey see Dytso et al. (2018). For different purposes, any of these parameterizations can be used to derive modified representations of the distributions considered in this paper.
Model extensions: a concluding remark
Although Definition 5 deals with the whole class of pspherical distributions later consideration is concentrated on continuous pspherical distributions. To finally widen again the view we refer to Remark 1 in Richter (2015a) where a way is described to derive new distributions from the elements of a given class of distributions by restricting the region of definition of such distributions. The following definition sums up that for the present situation.
Definition 8
The extension of the whole class of pspherical distributions follows accordingly.
It might be a further task of future work to extend the class of qspherical processes, q>0, introduced in Müller and Richter (2019) to a class of pspherical processes, p=(p_{1},...,p_{k})∈(0,∞)^{×k}.
Declarations
Acknowledgement
The author is grateful to the Referees for their valuable hints.
Funding
Not applicable.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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