The class of (p,q)-spherical distributions with an extension of the sector and circle number functions

Richter, Wolf-Dieter Article The class of (p,q)-spherical distributions with an extension of the sector and circle number functions Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Richter, Wolf-Dieter (2017) : The class of (p,q)-spherical distributions with an extension of the sector and circle number functions, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 5, Iss. 3, pp. 1-17, https://doi.org/10.3390/risks5030040 This Version is available at: https://hdl.handle.net/10419/195888 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article The Class of (p,q)-spherical Distributions with an Extension of the Sector and Circle Number Functions Wolf-Dieter Richter University of Rostock, Institute of Mathematics, Ulmenstraße 69, Haus 3, 18057 Rostock, Germany; wolf-dieter[email protected]; Tel.: +49-381-498-6551 Academic Editor: Mogens Steffensen Received: 24 May 2017; Accepted: 19 July 2017; Published: 21 July 2017 Abstract: For evaluating the probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based on the Gauss–Laplace law. The latter will be considered here as an element of the newly-introduced family of (p , q) -spherical distributions. Based on a suitably-defined non-Euclidean arc-length measure on (p , q) -circles, we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly to with elliptically-contoured distributions and more general homogeneous star-shaped ones. This is demonstrated by the generalization of the Box–Muller simulation method. In passing, we prove an extension of the sector and circle number functions. Keywords: Gauss-exponential distribution; Gauss–Laplace distribution; stochastic vector representation; geometric measure representation; (p , q) -generalized polar coordinates; (p , q) -arc length; dynamic intersection proportion function; (p , q) -generalized Box–Muller simulation method; (p,q)-spherical uniform distribution; dynamic geometric disintegration 1. Introduction The Gauss-exponential distribution plays an important role as a high risk limit law; see Sections 8 and 9 of the lectures presented in Balkema and Embrechts (2007) on high risk scenarios and extremes. Needless to recall here are the numerous different fields where quantitative risk management applies. The Gauss-exponential distribution can be considered as a particular asymmetric derivation of the Gauss–Laplace law. In particular, Gauss-exponential probabilities of arbitrary events can be dealt with by considering the corresponding Gauss–Laplace probabilities. Density level sets of the standard Gauss–Laplace distribution are topological boundaries of star bodies centered at the origin. The Minkowski functionals of the corresponding star bodies, however, are not homogeneous functions of order one, as is often assumed in the literature on star-shaped distributions. Instead, the bodies corresponding to different density levels reflect different geometric properties and are typically directed in different directions. The aim of the present paper is to model (p , q) -spherical generalizations of the two-dimensional Gauss–Laplace distribution. We prove geometric and stochastic representations, which can be considered as standard tools for dealing with the present distributions later on in a way similar to how one has already for a long time successfully been dealing with elliptically-contoured and, since more recently, even with more general homogeneous star-shaped distributions. This will be shortly indicated here by generalizing the Box–Muller simulation method. It is well known from two-dimensional spherical distribution theory that a random vector X following such distribution allows a stochastic representation Xd =R·U with independent non-negative Risks 2017,3, 40; doi:10.3390/risks5030040 www.mdpi.com/journal/risks Risks 2017,3, 40 2 of 17 random variable R and singular (with respect to the Lebesgue measure in R2 ) random vector U being uniformly distributed on the Euclidean unit circle. Understanding the suitable way of generalizing the latter distribution requires the most effort in general homogeneous star-shaped distribution theory. The corresponding singular distribution is dealt with by several authors (even in higher dimensions) by considering densities of marginal variables (vectors); see (Yue and Ma 1995) and (Song and Gupta 1997) for the particular case of lp -spherical distributions. Studying a disintegration formula for this case, it is proven in (Rachev and Rueschendorf 1991) that the geometric surface measure cannot coincide with their uniform distribution if p/∈ { 1, 2, ∞} . Moreover, the authors mention that ‘its treatment seems to need a completely different proof than the proof for the uniform distribution given in this (their) paper.’ In (Schechtman and Zinn 1990) , the authors exploit properties of the distribution being called the uniform distribution on lp -spheres in (Rachev and Rueschendorf 1991), by making use of a representation of it later called a cone measure representation; see (Barte et al. 2005). An insightful Kepler law interpretation of this measure is discussed in (Wallen 1995). A coordinate-based approach to describing uniform distributions on lp -spheres is given in (Szablowski 1998) where it is inter alia said that ‘it seems that the usage of the word uniform’ ... ’ does not refer to the real, geometrical uniformity of the probability mass on the surface of the unit sphere in n -dimensional Lα ’. The differential geometric explanation of the generalized uniform distribution given for the particular case of ln,p -spheres for arbitrary finite dimension in Richter (2009) (and for dimension two already in two earlier papers on the circle number function mentioned there) provided a qualitatively new approach to this long standing measure theoretical problem. Elliptically contoured and more general homogeneous star-shaped distributions are studied analogously in (Richter 2011a,2013,2014,2015a,2015b,2016a,2016b) based on the consideration of suitably-introduced non-Euclidean geometries. Much effort is expected to be necessary to give a suitable explanation of the geometric nature of a uniform distribution in the present case that the Minkowski functional of the density contour sets defining the star body is not homogeneous of order one. In going through all of the necessary steps to reach such an interpretation, we will be confronted with generalizing the notions of circle and its radius, as well as its circumference and with an extension of the sector and circle number functions. In the homogeneous case, that is if the mentioned Minkowski functional is homogeneous of order one, the analogous steps can be observed in Richter (2007) and in the author’s series of papers mentioned before. Even more information on the history of the mentioned measure theoretical problem can be found there. Among others, a technical key role will be played here by the suitable choice of coordinates for describing generalized uniform distributions. Moreover, we make a next basic step of extending the sector and circle number functions to classes of generalized circles having different geometric properties for different values of their generalized radii. The density of the standard Gauss–Laplace law Φ∗ G,Lin R2is given by: Φ∗ G,L(d(x,y)) = 1 2√2πe−x2 2−|y|d(x,y). Let us consider the (p,q)-spherical generalized normal density: Φp,q(d(x,y)) = CpCqe−|x|p p−|y|q qd(x,y) where p> 0, q> 0 and Cp=p1−1/p/( 2 Γ( 1 /p)) . We note that Φ2,1 =Φ∗ G,L and Φ2,2 and Φ1,1 are two-dimensional standard Gauss and Laplace distribution laws, respectively. Figure 1shows (p,q)-spherical densities for different choices of (p,q). Risks 2017,3, 40 3 of 17 0.01 0.005 0 -0.005 p=.2,q=.75 -0.01 -2 -1 0 1 0 5 10 2 1 2 3 4 5 6 7 8 Figure 1. (p,q)-spherical densities. Risks 2017,3, 40 4 of 17 The case p=q has been considered elsewhere, and we emphasize that this case will not be dealt with in the present paper. Thus, p6=qis assumed here. Let: |(x,y)|p,q=|x|p p+|x|q q,(x,y)T∈R2 denote the functional generating the density level sets or contour lines: Sp,q(r) = {(x,y)T∈R2:|(x,y)|p,q=r},r>0. Such a level set can be generated from the (p , q) -generalized unit circle Sp,q=Sp,q( 1 ) by matrix multiplication: Sp,q(r) = Dp,q(r)Sp,q and will be called the (p , q) -circle of (p , q) -radius r . Here, Dp,q(r) = diag(r1/p , r1/q) with diag(a , b) denoting the diagonal matrix with diagonal entries aand b. The star body: Kp,q(r) = {(x,y)T∈R2:|(x,y)|p,q≤r} having the origin in its interior and Sp,q(r) as its topological boundary will be called the (p , q) -circle disc of (p,q)-radius r. It satisfies the subset relation: Kp,q(r1)⊂Kp,q(r2)if r1<r2 and allows the representation: Kp,q(r) = r [ ρ=0 Sp,q(ρ). Such a disc is convex if p≥ 1 and q≥ 1 and is radially concave if 0 <p≤ 1 and 0 <q≤ 1. For the latter notion, we refer to (Moszy´nska and Richter 2012). Figure 2(drawn with MATLAB, as Figure 1) shows ( 2, 1 ) -circles S2,1(r) according to different values of the ( 2, 1 ) -generalized radius r , starting from a local (central) and turning to more global view. Roughly said, the impression of the main orientation of density level lines is changing within two steps from ‘west-east’ to ‘south-north’. The paper is organized as follows. Section 2presents preliminary material on (p , q) -generalizations of the common polar coordinates, the notions of arc-length and geometric disintegration, the sector and circle number functions and the generalized uniform distribution on a generalized circle. After further developing the general methodology from the theory of homogeneous star-shaped distributions, we are in a position to formally introduce the family of (p , q) -spherical distributions in Section 3 including the (p , q) -generalized normal distributions as particular cases of such distributions having a density. Section 4deals with a geometric generalization of the asymmetric Gauss-exponential law, and Section 5is devoted to some aspects of simulation. The discussion in Section 6delivers a look back to and ahead toward the present distribution theory. Finally, we give some conclusions in Section 7. Risks 2017,3, 40 5 of 17 p=2,q=1 -0.15 -0.1 -0.05 0 0.05 0.1 -0.15 -0.1 -0.05 0 0.05 0.1 0 0.05 0.1 0.15 p=2,q=1 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 0 1 2 3 4 5 6 7 p=2,q=1 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 0 5 10 15 20 Figure 2. Gauss–Laplace density level sets: from local (central) to global view. Risks 2017,3, 40 6 of 17 2. Preliminaries 2.1. A Class of (p,q)-Generalized Polar Coordinates Let Np(φ)=(|cos φ|p+|sin φ|p)1/p . The p -generalized cosine and sine functions are defined according to Richter (2007) by: cosp(φ) = cos φ Np(φ)and sinp(φ) = sin φ Np(φ). Definition 1. The (p,q)-generalized polar, or (p,q)-spherical, coordinate transformation: Polp,q:(0, ∞)×[0, 2π)→R2\{(0, 0)T} is defined by (x,y)T=Polp,q(r,φ)where: x= (pr)1/p(cospq(φ))q, 0 ≤φ<π/2, 3π/2 ≤φ<2π x=−(pr)1/p(−cospq(φ))q,π/2 ≤φ<3π/2 y= (qr)1/q(sinpq(φ))p, 0 ≤φ<π y=−(qr)1/q(−sinpq(φ))p,π≤φ<2π. Lemma 1. The absolute value of the Jacobian of this transformation is: J(Polp,q)(r,φ) = r1/p+1/q−1J∗(φ) where: J∗(φ) = p1/pq1/q|cospq(φ)|q−1|sinpq(φ)|p−1Npq(φ)−2. (1) The proof of this lemma makes use of the results in Richter (2007). Moreover, the inverse of the coordinate transformation Polp,qis given by: r=|(x,y)|p,q, φ=       π 2if y>0, x=0 arctan (p/q)1/(pq)y1/p x1/q!if y>0, x6=0 and: φ=         3 2πif y<0, x=0 arctan −(p/q)1/(pq)(−y)1/p x1/q!if y<0, x6=0. 2.2. The (p,q)-arc Length Measure and Dynamic Geometric Disintegration of the Lebesgue Measure In this section, first, the (p , q) -generalized arc length and the area content of (p , q) -circles and -circle discs are introduced, respectively. Then, a new type of geometric disintegration of the Lebesgue measure will be established. In the next section, extensions of the sector and circle number functions are Risks 2017,3, 40 7 of 17 considered. To start with, we recall that the area content of the (p , q) -circle disc with (p , q) -generalized radius ρis: µ(Kp,q(ρ)) = Z Kp,q(ρ) d(x,y) = ρ Z 0 2π Z 0|J(Polp,q)(r,φ)|dφdr. Successively changing variables x=cospq(φ)and y=xpq shows that: µ(Kp,q(ρ)) = 4B(1 p,1 q)p1 pq1 q p+qρ 1 p+1 q. (2) We note that the power exponent of ρ reflects a certain aspect of the change of shape of Kp,q(ρ) when ρ varies. Moreover, we emphasize the remarkable differences between convex and radially-concave cases. In the sequel, in a certain analogy to what was done in ( Richter 2007,2009,2011b,2013 )for various situations, we define the (p , q) -generalized arc length measure ALp,q on the Borel σ -field B(Sp,q) of Sp,q . To this end, for an arbitrary set A∈ B(Sp,q) , let us call: CPCp,q(A) = {Dp,q(r)(x,y)T:(x,y)T∈A,r>0} a nonlinear matrix transformed central projection cone. Note that |x|p/p+|y|q/q= 1 if (x , y)T∈A and that Dp,q(r1)(x , y)T6=Dp,q(r2)(x , y)T if r16=r2 . The set CPCp,q(A) can be considered as a union of pairwise disjoint sets, CPCp,q(A) = [ r>0 [Dp,q(r)A]. Here, multiplication of a set by a matrix is defined in the common pointwise sense. Furthermore, a (p,q)-sector of Kp,q(r)having (p,q)-generalized radius ris defined by: Sep,q(A,r) = CPCp,q(A)∩Kp,q(r), and we consider the (p,q)-radius dependent area content function: f(ρ) = µ(Sep,q(A,ρ)) = ρ Z 0Z Pol∗−1 p,q(A) |J(Polp,q)(r,φ)|d(r,φ),ρ≥0. (3) Here, Pol∗ p,q(φ) = Polp,q(1, φ), and Pol∗−1 p,qdenotes the inverse function of Pol∗ p,q. Definition 2. The (p,q)-arc length measure ALp,q(A)is defined for arbitrary A ∈ B(Sp,q)by: ALp,q(A) = f0(1). It follows from the definition of fthat: ALp,q(A) = Z Pol∗−1 p,q(A) |J∗(φ)|dφ,A∈ B(Sp,q)(4) and: ALp,q(Dp,q(r)A) = r1/p+1/q−1ALp,q(A). (5) In particular, ALp,q(Sp,q(r)) = 4B(1 p,1 q)p1/p−1q1/q−1r1/p+1/q−1. (6) Risks 2017,3, 40 8 of 17 If B∈ B2has finite area content, then: µ(B) = Z B dx = ∞ Z 0 2π Z 0 IB(Polp,q(r,φ))|J(r,φ)|dφdr = ∞ Z 0 r1/p+1/q−1 2π Z 0 IB(Polp,q(r,φ))|J∗(φ)|dφ dr. Because IB(Polp,q(r,φ)) = 1 if and only if Polp,q(r,φ)∈Sp,q(r)∩B, it follows that: µ(B) = ALp,q(Sp,q) ∞ Z 0 r1/p+1/q−1Fp,q(B,r)dr (7) where: Fp,q(B,r) = ALp,q([Dp,q(r−1)B]∩Sp,q)/ALp,q(Sp,q),r>0 (8) denotes the (p , q) -spherical intersection proportion function of the set B . By similar argumentation, the following theorem is proven. We note that (7) represents a new non-Euclidean type of geometric disintegration of the Lebesgue measure in R2 , which due to the effects of action Dp,q will be called a dynamic disintegration. For this reason, the function in (8) will be called a dynamic intersection proportion function, and the integration method in (7) will be called dynamic geometric disintegration of the Lebesgue measure. Theorem 1. If h is integrable over B, then the dynamic geometric disintegration: Z B h(x)dx = ∞ Z 0   r1/p+1/q−1Z B∗(r) h(Polp,q(r,φ))|J∗(φ)|dφ  dr is valid where: B∗(r) = {φ∈[0, 2π):Polp,q(1, φ)∈[Dp,q(r−1)B]∩Sp,q}. 2.3. The (p,q)-Circle and Sector Number Functions Circle numbers of star discs are studied in Richter (2011b) and for particular cases in earlier papers cited therein. Similarly, we define the (p , q) -circle number function to assign the number πp,q to any (p,q)-circle of (p,q)-radius rwhere for any r>0: µ(Kp,q(r)) r1/p+1/q=πp,q=ALp,q(Sp,q(r)) (1/p+1/q)r1/p+1/q−1. As for any A∈ B(Sp,q), f0(r) = ALp,q(Dp,q(r)A), we can also define the (p , q) -sector number function to assign the number πp,q(A) to any (p , q) -sector Sep,q(A,r)of (p,q)-radius rwhere: µ(Sep,q(A,r)) r1/p+1/q=πp,q(A) = ALp,q(Dp,q(r)A) (1/p+1/q)r1/p+1/q−1,∀r>0 Risks 2017,3, 40 15 of 17 By (13), the density of angle Φis given as: fΦ(φ) = pq B(1/p, 1/q)|cospq(φ)|q−1|sinpq(φ)|p−1 N2 pq(φ), 0 ≤φ<2π. (18) Starting from this representation, one can proceed as described in (Kalke and Richter 2013) and Richter (2015a), Example 9(b), or any of the standard monographs on simulation mentioned there. 6. Discussion The way of probabilistic modeling developed in this paper is closely related to various challenging mathematical problems. It is well known from the results in (Richter 2014,2016a,2016b) and the references given there that representations of star-shaped distributions whose contour-defining star body has a homogeneous Minkowski functional of order one are closely related to suitably-chosen non-Euclidean geometries. Here, we discover that there is again a need to go some steps beyond such geometries and realize the first of them. Already in the 17th Century, basically starting from the work of Descartes, various coordinate systems played a fruitful role in geometric applications. Nevertheless, it seems that suitably chosen coordinates may serve even these days as a powerful tool for solving nontrivial problems in different areas of mathematics. In the present case, star bodies whose Minkowski functionals are not homogeneous functions of degree one are effectively described for the purposes of representing two-dimensional Gauss–Laplace laws and their (p , q) -spherical generalizations with the help of generalized polar coordinates based on generalized sine and cosine functions. Starting latest from the work of Leibniz and Newton who founded modern calculus, in many areas of mathematics, one makes use of thin parallel layers when defining and studying certain basic notions. Here, however, small changes of a generalized radius variable related to such a body generate thin layers close to the bodies’ boundary, being nonparallel. To the best of the author’s knowledge, the fundamental measure theoretical problem of understanding the factorization components of cross-sections or disintegrations of the present type seems to be approached here for the first time. The present work extends the line of interchanging the role that the notions of circle and distance play in comparison with Euclidean geometry, described inter alia in (Richter 2011a,2011b). Here, the ‘circle’ is given by a density level set modeling a ‘contour line’ of a sample cloud, and the understanding of what is a ‘distance’ leads to a directionally-dependent notion of radius being related to a matrix-vector multiplication. This remains, however, that the question of what is the differential geometric meaning of the newly-introduced (p , q) -generalized arc length measure. Therefore, it is stated here as an open problem. Finally, we remark that the results in Section 2allow the following additional representations of the Lebesgue measure, which may be useful in future applications of (p , q) -spherical distributions. For r∈(0, 1),φ∈[0, 2π), µ(dx) = (1/p+1/q)r1/p+1/q−1dr p1/pq1/q 1/p+1/q|cospq(φ)|q−1|sinpq(φ)|p−1dφ N2 pq(φ) and for ρ∈(0, 1], µ(Sep,q(A,ρ)) = Zρ 0ALp,q(Dp,q(r)A)dr. An alternative representation is given for r∈(0, 1),t∈(0, 1)by: µ(dx) = (1/p+1/q)r1/p+1/q−1dr t1/p−1(1−t)1/q−1dt. Risks 2017,3, 40 16 of 17 7. Conclusions The Gauss-exponential distribution being of particular interest in high risk scenarios can be numerically dealt with now based upon a method newly developed in this paper. This method, moreover, opens new perspectives for studying in the future broad classes of multivariate distributions not just being homogeneous of order one. A detailed and full developement of this distribution theory will further bring together methods at least from measure theory, non-Euclidean differential geometry, isoperimetry, extending ball and sector number functions, defining suitable coordinates, solving partial differential equations, and functional analysis. Conflicts of Interest: The author declares no conflict of interest. References Balkema, Guus, and Paul Embrechts. 2007. High Risk Scenarios and Extremes. 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