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Highdimensional starshaped distributions
Journal of Statistical Distributions and Applications volume 6, Article number: 5 (2019)
Abstract
Stochastic representations of starshaped distributed random vectors having heavy or light tail density generating function g are studied for increasing dimensions along with corresponding geometric measure representations. Intervals are considered where star radius variables take values with high probability, and the derivation of values of distribution functions of grobust statistics is proved to be based upon considering random events whose probability is asymptotically negligible if the dimension of the sample vector is approaching infinity. Moreover, a principal component representation of pgeneralized elliptically contoured pgeneralized Gaussian distributions is discussed.
Introduction
Among the frequently obtained impressions one gets from analyzing highdimensional data sets are that an observation point’s distance from the zero element of the sample space is likely to belong to a certain interval from the positive real line, away from zero, and that the distribution of the direction of the vector seems to be close, in a certain sense, to a uniform distribution on the set of all directions that are observable from a certain center. The first observation can be reflected from a probabilistic point of view by a measure concentration type property including what is done in (Biau and Mason 2015) and (Vershynin 2016), and the second is part of background for testing uniformity on highdimensional spheres, see e.g. (Cutting et al. 2017), possibly after projecting data points onto spheres as, e.g., in (Banerjee and Ghosh 2004).
In situations of the described type, it may be reasonable to model the data, or their residuals after fitting to a model, by multivariate starshaped distributions. In this regard, (Balkema and Embrechts 2007) and (Balkema et al. 2010) discover conditions ensuring that starshaped distributions with the Gaussexponential law being one of the most known examples appear as limit laws in certain highrisk scenarios.
Distributions from the class of starshaped distributions are flexible with respect to convexity or radial concavity, allow different variability of probability mass along different directions of the sample space and are able to model light and heavy distribution centers and tails. Because there is no natural number being representative for large dimensions, one might like to consider sequences or schemas of series of ndimensional vectors with n approaching infinity. However, for simplicity of notation, we instead consider here just a single random vector X taking values in \(\mathbb {R}^{n}\) and assume afterwards that n is tending to infinity in formulas holding for X.
Let us recall at this point the following general aspect of uni or multivariate asymptotic probabilistic analysis being of particular importance, for example, in large deviation theory, but not exclusively there. Studying the limit behavior of certain sequences of distributions on specific subsets of their ranges of definition and comparing it to how the appearing limit law itself behaves on the same sets needs to precisely know the latter one. In this respect, it is an independent problem to study the behavior limit laws show on the sets of interest. Similarly, if a sequence of distributions of increasing dimension is approximated in a certain part of its range of definition by a highdimensional starshaped limit law then studying the latter one is an independent problem being in the core of interest of the present note.
With the agreement of considering just one single vector X of dimension n, particular questions concerned by the buzzword ’big data’ are approached in the present short note by reflecting above mentioned impressions gained from data in the language of probability distributions. To be more specific, we are dealing here with starshaped distributions in \(\mathbb {R}^{n}\) and correspondingly distributed vectors. Such vector allows a stochastic representation as a product of a random generalized radius variable R and a random vector U being staruniformly distributed on a starsphere and independent of R, as well as a corresponding geometric measure representation. Some consequences which can be drawn from these representations in case of increasing dimension are studied. In particular, a representation of pgeneralized elliptically contoured distributions is considered from the point of view of principal components.
The paper is structured as follows. In “Preliminaries” section, we present preliminary facts on starshaped distributions including the notions of star surface content measure and staruniform distribution on a star sphere. “A principal component representation” section deals with the particular class of pgeneralized elliptically contoured distributions and it is studied there how they apply to modeling highdimensional data. “A measure concentration property” section is then aimed to consider typical intervals where R takes values if X is starshaped distributed, and in “On grobust statistics” section distributions of univariate statistics are described which can basically be derived from staruniformly distributed vectors. Such distributions are not affected by whether X has a density generating function g generating light or heavy distribution tails and is therefore called grobust. The derivation of values of distribution functions of grobust statistics is proved to be based upon considering random events whose probability is asymptotically negligible if the dimension of the sample vector is approaching infinity.
Preliminaries
Let \(K\subset \mathbb {R}^{n}\) be a star body having the origin in its interior and assume that the Minkowski functional h_{K} of K is positively homogeneous of degree one. We call K(r)=rK and its boundary S(r)=rS the star ball and star sphere of star radius r>0, respectively. If g:[0,∞)→[0,∞) satisfies 0<I(n,g)<∞ where \(I(n,g)=\int \limits _{0}^{\infty } r^{n1}g(r)dr\) then it is called a density generating function. In such case,
is called a starshaped density and K its contour defining star body. The corresponding probability measure is denoted Φ_{g,K} and the normalizing constant allows the representation
where \(\mathfrak {O}_{S}(S)\) means the stargeneralized surface content of S, see (Richter 2014). If the additional assumption C(g,K)=1 is satisfied then g is called a density generator. In the following example, we recall an explicit analytical representation of the stargeneralized surface content measure O_{S} in case S is a pgeneralized ellipsoid with main axes of half lengths a_{1},...,a_{n} and indicate relationships to representations in other particular cases.
Example 1
Let \(K=\{x\in \mathbb {R}^{n}:x_{a,p}\leq 1\}\) where \(x_{a,p}=\left (\sum \limits _{i=1}^{n}\frac { x_{i}}{a_{i}}^{p}\right)^{1/p}, a=(a_{1},...,a_{n})^{T}, a_{i}>0,i=1,...,n,p>0\), and S^{+(−)}=S∩{x:x_{n}>(<)0} the upper (lower) half of the (a,p)ellipsoid S={x:x_{a,p}=1}. Then
where \(G(A\cap S^{+()})=\{\vartheta \in \mathbb {R}^{n1}:\exists \eta =\eta (\vartheta) s.t. (\vartheta ^{T},\eta)^{T}\in A\cap S^{+()} \}\), \(\mathfrak {B}(S)=\mathfrak {B}^{n}\cap S\) and \(\mathfrak {B}^{n}\) denotes the Borel σfield in \(\mathbb {R}^{n}\).

If p=2 and a=1_{n}=(1,...,1)^{T} then \(\mathfrak {O}_{S}(A)\) is the Euclidean surface content of the measurable subset A of S.

If p=1 then \(\mathfrak {O}_{S}(A)\) can be considered as a particular polyhedral generalized surface content of A.

If \(\mathfrak {O}_{S,\infty }(A)\) is defined as the limit of \(\mathfrak {O}_{S}(A), A\in {\mathfrak {B}(S)}\) as p→∞ then \(\mathfrak {O}_{S,\infty }\) can be considered as another particular polyhedral generalized surface content measure. For the whole class of polyhedral generalized surface content measures, see (Richter and Schicker 2017).

Generalizations of representation (1) hold true for all cases where K is a ball with respect to any norm or antinorm, see (Richter 2015).
The next example deals with the asymptotic behavior of star surface content and volume of star spheres and star balls or ellipsoids, respectively, if dimension is approaching infinity.
Example 2
[a] It is well known that if S is the Euclidean unit sphere then \(\mathfrak {O}_{S}(S)=\omega _{n}\) where \( \omega _{n}={2\pi ^{n/2}}/{\Gamma ({\frac {n}{2}})} \) is the Euclidean surface content of S. It is known that \(\arg \sup \limits _{n}\omega _{n}=7\) and that ω_{n} is monotonously decreasing starting from this value, see e.g. (Loskot and Beaulieu 2007). Moreover, according to Stirling’s formula,
meaning that the ratio of the quantity on the left hand side divided by that of the right hand side tends to one if n tends to infinity. Obviously, \(\mathfrak {O}_{S}(S)\) is tending to zero quite fast as n→∞.
(b) If S is the l_{n,p}sphere having unit star radius, p>0, then the star surface content of S is known to be \(\mathfrak {O}_{S}(S)=\omega _{n,p}\) where \( \omega _{n,p}={2^{n}(\Gamma (\frac {1}{p}))^{n}}/(p^{n1}\Gamma (\frac {n}{p})). \) Note that
Let \(\Omega _{n,p}=\frac {\omega _{n,p}}{n}\) denote the volume of the l_{n,p}ball. The asymptotic relations following from the latter one,
generalize two results given in (Chen and Lin 2014) for the particular Euclidean case p=2.
(c) It is well known that the star surface content of the (a,p)ellipsoid \(S=\{x\in \mathbb {R}^{n}:x_{a,p}= 1\}\) is \( \mathfrak {O}_{S}(S)=a_{1}\cdots a_{n}\omega _{n,p}, \) thus
If a random vector X follows the starshaped density φ_{g,K} then it allows the stochastic representation
meaning that X is distributed as R·U. The nonnegative random variable R is independent of the random vector U, R has density function
and U has the staruniform distribution
Here, I_{[0,∞)}(r)=1 if r≥0 and I_{[0,∞)}(r)=0 otherwise. Accordingly, the geometric measure representation of starshaped distribution laws reads
where
is the star sphere intersection proportion function of the set B. Let the Minkowski functional of K be denoted h_{K} then
and R is called the star radius of X.
A principal component representation
In this section we study to what extent a particular class of continuous multivariate starshaped distributions applies to modeling highdimensional data. To be specific, we consider pgeneralized elliptically contoured pgeneralized Gaussian distributions.
It will be shown which way principal component analysis can be used to identify those components of such starshaped distributed vectors being of major importance for the modeling process. In particular it turns out that covariance ellipsoids being l_{2}ellipsoids have the same main axes as density level set ellipsoids being l_{p}ellipsoids.
Let X_{i},i=1,...,n be independent random variables correspondingly following the densities
where \(C_{p}=p^{11/p}/\left (2\Gamma \left (\frac {1}{p}\right)\right)\). Then \(\mathbb {E}X_{i}=\mu _{i}\) and \(V(X_{i})=\kappa \sigma _{i}^{2}\) are expectation and variance of X_{i}, respectively, i=1,...,n. Here and in what follows we use the notation
Let σ=(σ_{1},...,σ_{n})^{T},X=(X_{1},...,X_{n})^{T} and μ=(μ_{1},...,μ_{n})^{T}, then X−μ allows the stochastic representation
where R_{σ,p}=X−μ_{σ,p} and \(U_{\sigma,p}=\frac {1}{R_{\sigma,p}}(X\mu)\) are independent. The star radius R_{σ,p} follows the pgeneralized Chidistribution with n d.f. having according to (Richter 2007) the density
and the stochastic basis vector U_{σ,p} is staruniformly distributed on the star sphere S=E_{σ,p} from Example 2(c) with a_{i}=σ_{i},i=1,...,n,U_{σ,p}∼ω_{σ,p}. Thus, using notation in (Richter 2014), X belongs to the class \( EC_{\sigma,p,\mu,I_{n}}\) and its density generating function can be chosen as \(g(r)=g_{p}(r)= e^{\frac {r^{p}}{p}}I_{[0,\infty)}(r)\). Note that \(\mathbb {E} U_{\sigma,p}=0_{n}\) and, because \(\mathbb {E} R^{2}_{\sigma,p}=\kappa (n)\),
where
It follows that \(\mathbb {E} X=\mu \) and cov(X)=κD^{2}. Now, denote an orthogonal n×nmatrix \(O=\left (O_{i,j}\right)_{i,j=\overline {1,n}}\) and let the transpose of its i’th row be O_{i}=(O_{i,1},...,O_{si,n})^{T}. The random vector Y=O(X−μ) follows the pgeneralized elliptically contoured density
that is Y belongs to the class \(EC_{\sigma,p,0_{n},O}. \) Note that \(\mathbb {E}Y=0_{n}\) and \(cov(Y)=\mathbb {E}YY^{T}=\Sigma =\kappa OD^{2}O^{T}=\left (\sigma _{i,j}\right)_{i,j=\overline {1,n}}\) where
Thus, \(\phantom {\dot {i}\!}f_{Y}=\varphi _{g_{p},OK}\) and the boundary of K is S=E_{σ,p}. Note that OK is a star body having the properties introduced at the beginning of this section. The covariance ellipsoid of Y is
where Π_{y}x means the orthogonal projection of x into the linear space spanned up by y. The main axes of C(Σ) belong to the spaces spanned up by the vectors O_{i} and have half lengths of size \(\sqrt {\kappa }\sigma _{i}, i=1,...,n\), respectively. Moreover, the set C(Σ) is symmetric with respect to any of the lines \(L_{i}=\mathfrak {L}\{O_{i}\}, i=1,...,n.\) The latter holds also true for the pgeneralized ellipsoids
because E_{σ,p} is symmetric with respect to the lines
for i=1,...,n, and
with e_{1},...,e_{n} being the standard orthonormal basis vectors in \(\mathbb {R}^{n}\). One may chose the variances according to the maximization procedure from principal component analysis, σ_{1}≥σ_{2}≥...≥σ_{n}. For highdimensional data holds σ_{n}→0 as n→∞. This circumstance allows to introduce data reduction. Therefore, methods from this area of statistical analysis apply to model data of arbitrary fixed or increasing dimension by pgeneralized elliptically contoured distributions if p≥1. If p∈(0,1), certain maximization principles from PCA are to be changed with corresponding minimization principles. In this sense, representation (3) may be called a principal component representation of pgeneralized elliptically contoured pgeneralized Gaussian densities. With slight changes, the density generating function g=g_{p} may be replaced in (3) with an arbitrary one for representing general pgeneralized elliptically contoured densities.
A measure concentration property
A χ^{2}distributed random variable with n d.f. can be considered as the square of the Euclidean norm of an ndimensional standard Gaussian vector. The probability mass of such vector is the more concentrated in a relatively small shell having radius of order \(\sqrt {n}\) the larger the vector’s dimension is, see Figs. 1 and 2. Observing the empirical ”relative concentration numbers” 30/10,45/30,90/100 and 300/1000 one may argue that suitably defined numbers might even converge to zero in some sense. This will be proved here within the even more general frame of χ^{p}distributions. For definitions and properties of these distributions we refer to (Richter 2007; 2009; 2014; 2015; 2016).
If a multivariate distribution converges to the standard Gaussian law then the square of the Euclidean norm of the correspondingly distributed vector X, i.e. the square of the Euclidean radius R=X_{1,2} of such vector, will tend under some additional assumption to the χ^{2}distribution with n d.f.. Let now X follow a starshaped distribution Φ_{g,K}, what can we say then about the behavior of (the suitably defined power of) its generalized (star) radius? In this section we derive typical intervals where star radius variables of highdimensional starshaped vectors take values.
Proposition 1
For δ>0 chosen such that
is approaching zero as dimension n is tending to infinity, and independently of the shape defining star body K, the typical behavior of the random star radius of a vector following the starshaped distribution Φ_{g,K} with density generating function g is described by
Proof
Obviously,
Now, Tschebyscheff’s inequality applies \(\square \) □
According to (Biau and Mason 2015), the behavior of X_{1,p} as n increases is called the distance concentration phenomenon in the computational learning literature. For sums of independent random variables or matrices, sharper concentration inequalities of exponential type are proved in (Vershynin 2016).
For more details on moments of pspherical random vectors, see (ArellanoValle and Richter 2012), for an asymmetric situation if p=1 see (Henschel and Richter 2002). The following corollary deals with a class of light tailed highdimensional starshaped distributions.
Corollary 1
Let K be any star body as introduced in “Preliminaries” section. If the density generating function of a highdimensional starshaped distribution Φ_{g,K} is that of Kotz type with parameters s>0,t>0,k>1−n,
and
(meaning that \(\delta {\sqrt {n} }\rightarrow \infty \) as n→∞) then, for sufficiently large n, there holds
Proof
First we check that α tends to zero: because
where O(.) means Landau’s big O symbol, it follows
Such α approaches zero as n tends to infinity if δ^{2}·n→∞ for n→∞. Finishing the proof, we finally observe that
□
Remark 1
On using Stirling’s formula, it can be seen that
Before turning to the case of heavy distribution tails, we note that asymptotic relation (4) makes the statement of Corollary 1 more specific in the sense that
is a reasonable interval where R takes values with high probability if we are given a highdimensional starshaped vector with density generating function of Kotz type and fixed or increasing dimension n. The most essential role is played here by parameter s which basically determines the relative heaviness or lightness of the distribution tails.
Corollary 2
Let K be any star body as introduced in “Preliminaries” section. If the density generating function of a starshaped distribution Φ_{g,K} is that of Pearson type VII with parameters s>0,k>n+2 [12] where k−n→∞ as n→∞,
and
then, for sufficiently large n, with probability greater or equal to 1−α, the starradius R of the highdimensional vector X belongs to the interval
Proof
Checking that α tends to zero, we find that
The proof is finished by observing that
□
Remark 2
Let δ=δ(n)→+0 and α=α(n)→+0 as n→∞ such that nδ^{2}(n)→∞ and assume that in the situation of Corollary 2 there holds additionally that (k−n)δ^{2}(n)→∞. The statements of Corollaries 1 and 2 can be reformulated then as
where Landau’s symbol O defined for the asymptotic relation f(n)=O(g(n)),n→∞ guaranties the existence of a constant C such that for all n there holds f(n)/g(n)≤C. Moreover, in the situation of Corollaries 1 and 2 we have that
respectively.
Remark 3
Let us finally mention that because in fact we are considering sequences or even schemas of series of vectors and distributions, the assumption k−n→∞ stated in Corollary 2 is not contradictory. Instead, it ensures a certain variability of the result. Moreover, we remark that (Henschel 2001) and (Henschel and Richter 2002) study the exact distribution of R in case of simplicially contoured vectors (or l_{n,1}spherical vectors having nonnegative components). General l_{n,p}spherical vectors and their star radius R are studied in (Richter 2009), (ArellanoValle and Richter 2012) and (Richter 2014), tables of corresponding exact quantiles of R^{p} and R are to be found in (Müller and Richter 2016) and (Richter 2016).
On grobust statistics
If the distribution of a statistic does not depend on the density generating function g of a starshaped sample vector density φ_{g,K} then it is commonly called grobust. It is well known that Student and Fisher type statistics possess besides the grobustness property further optimality properties. Here we will see that decisions based upon such statistics are done by closer analyzing random events whose probability is asymptotically negligible if the dimension of the sample vector is approaching infinity.
To be more concrete, in this section, we describe a class of statistical distribution functions, derived from a starshaped sample vector, the ratio representations of whose values are asymptotically negligible as vector dimension increases unboundedly. It turns out, e.g., that classical and generalized Student and Fisher distributions belong to this class.
Let a random vector X follow the distribution law Φ_{g,K},X∼Φ_{g,K}, and the sets
being generated by a statistic \(T:\mathbb {R}^{n}\rightarrow \mathbb {R}\) such that the equation
is satisfied where \(\mathcal {C}(t)\in [0,1]\) does not depend on r. A statistic T of this type is grobust. It follows by the geometric representation proved for starshaped distributions in (Richter 2014) that
The distribution law of T(X) is determined already by that of vector’s X central projection onto the star sphere S. Thus, for \(t\in \mathbb {R}\), the probability P(T(X)<t) is defined by the ω_{S}value of the random event B(t)∩S in the geometric probability space \((S,\mathfrak {B}(S),\omega _{S})\), and its representation in (5) will be called its ratio representation. It can be observed that many star spheres show the asymptotic behaviour
For a particular case of such type, see Example 2(a). The statistical model concerned in this case is dealing with independent and homoscedastic random variables. In case of increasing dimensions, we are confronted then with sequences of probability spaces with asymptotically negligible set S.
Example 3
In case S is the unit l_{n,p}sphere, condition (6) is satisfied.
In the following example we restrict our consideration to the ndimensional standard Gaussian law Φ_{g,K}=Φ where \(g(r)=e^{r^{2}/2}\) and \(K=\left \{x: x_{1}^{2}+...+x_{n}^{2}\leq 1\right \}.\)
Example 4
A set A⊂R^{n} belongs to the class \(\mathfrak {A}(dir,dist)\) if there exist functions e_{A}: [0,∞)→S_{n}(1) and R_{A}: [0,∞)→[0,∞) satisfying the following two assumptions:
\(\mathfrak {A} 1)\)The set A allows the representation
where H_{n}(e,R)={x∈R^{n}:Π_{e}x=λe,λ≥R} is a half space and S_{n}(r) the Euclidean sphere of radius r.
\(\mathfrak {A} 2)\) The function \(\mathfrak {C}:[0,\infty)\rightarrow R^{n}\) with
is a piecewise continuous curve such that A becomes a Borel set.
The functions e_{A} and R_{A} are called directional type and distance type functions of the set A, respectively. If \(A\in \mathfrak {A}(dir,dist)\) then
where C(n)=2^{1−n/2}/Γ(n/2), see (Richter 1995). If the function r→α^{∗}(r)is constant then
where
Example 5
Let \(\frac {1}{\sigma }X,\sigma >0,\) be a standard Gaussian distributed random vector in R^{n}. The statistic
is known to be Student distributed with k d.f. for all \(\mathfrak {e}\in S_{n}(1)\) and all kdimensional linear subspaces \(\mathcal {N}\) of R^{n} such that \(\mathfrak {e}\bot \mathcal {N}\) and k≤n−1.
Let \(A=B(t)=\{T_{\mathfrak {e},\mathcal {N}}< t\}\). Then \(A\in \mathfrak {A}(dir,dist)\) where \(e_{A}(r)=\mathfrak {e},\) the distance type function is \( R_{B(t)}(r)=\tilde {t}r /(\tilde {t}^{\,2}+1)^{1/2},\ \tilde {t}=t/\sqrt {n1}\) and the function \( \alpha ^{*}(r)=\arctan \ (1/\tilde {t}\,) \)is constant. Evaluating with k=n−1 the limit of Φ(A)as n→∞ leads to the well known result Φ_{0,1}(t) where Φ_{0,1} denotes the cumulative distribution function of the univariate standard normal distribution.
For similar properties of a corresponding exact Student test in nonlinear regression, see (Ittrich 2000) and (Ittrich and Richter 2005).
Example 6
For a related consideration on the pgeneralized Fisher statistic, see (Richter 2009).
Remark 4
If one is interested in avoiding the asymptotic negligibility of S in case of increasing dimension one may leave the class of statistical models dealing with independent homoscedastic observations. Density level sets of sample vectors having heteroscedastic components may be starshaped. If S is an (a,p)ellipsoid then, according to asymptotic relation (2), condition (6) may by violated and even asymptotically stabilizing.
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Richter, WD. Highdimensional starshaped distributions. J Stat Distrib App 6, 5 (2019). https://doi.org/10.1186/s4048801900960
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Keywords
 Increasing dimension
 Starshaped distribution
 Star radius distribution
 Staruniform distribution
 pgeneralized elliptically contoured distribution
 Principal component representation
 grobust statistic
 Indeterminate form