Constructing models for spherical and elliptical densities

1 Introduction

Apart from copulas, elliptically contoured distributions (abbreviated: elliptical distributions) play an important role in modelling multivariate distributions. Models for elliptical distributions are applied in various fields, such as finance (see McNeil et al. [13], for example), ecological science, econometry, engineering, and other disciplines. The classical theory is presented in the popular textbooks by Anderson [2], Fang and Zhang [6], and Fang et al. [5]. Fundamental properties of elliptical distributions were published in the papers by Kelker [9], Cambanis et al. [3], and Kano [8], among others. Kano [8] introduced the term of consistent spherical distributions. Since the publication of the Kano article, it is an open problem whether there are principles for constructing consistent families of spherical distributions. Based on such a family, a corresponding family of elliptical distributions can be established. In this article, we show that the consistency of families of spherical distributions is closely related with complete monotonicity of generator functions. Contrary to most of the other papers, we define the continuous elliptical distribution by use of the density instead of characteristic functions.

Parametric estimation problems in the context of elliptical distributions were examined by a series of authors, for example, see the studies by Maruyama and Seo [12] and Srivastava and Bilodeau [17]. Semiparametric estimators for elliptical distributions were considered in the paper by Liebscher [10], where the generator function is estimated by a kernel estimator.

The classes of distributions that we focus on in this article are introduced next.

DEFINITIONS: A p -dimensional random vector X follows a continuous spherical distribution SC p ( g p ) if its density function exists and is given by g p ( t T t ) for t ∈ R p with a function g p : [ 0 , ∞ ) → [ 0 , ∞ ) . We write X ∼ SC p ( g p ) .

Function g p is referred to as a density generator for dimension p if g p is non-negative, and fulfills

A random vector Y ∈ R p has a continuous elliptical distribution (of full rank) with parameters μ ∈ R p and Σ ∈ R p , p , rank ( Σ ) = p , if Y has the same distribution as μ + A T Z , where Z ∼ SC p ( g p ) and A ∈ R p , p is a matrix such that A T A = Σ . In symbols Y ∼ EC p ( μ ; Σ ; g p ) .

For the density f Y of Y , we have

for y ∈ R p . Let s p = π p ∕ 2 Γ ( p ∕ 2 ) . Condition (1.1) ensures that t ↦ g p ( t T t ) ( t ∈ R p ) and f Y according to (1.2) represent a density. Next, we provide the theorems about the representation of spherical and elliptical random vectors.

From (1.4), it follows that

is the density of R 2 , where R is the random radius in (1.3). Considering specific families of generator functions, this formula will be used in the generation of spherical random variates according to Equation (1.3). Theorem 1.2 is a consequence of Theorem 1.1 (cf. [6], Corollary 1 to Theorem 2.6.1):

Unfortunately, the parametrization ( μ ; Σ ; g p ) of EC p densities is not unique. This is stated in Theorem 1.3 (cf. [6], Theorem 2.6.2):

Theorem 1.3 is the reason that we do not consider the scale parametrization of g p .

In this article, the aim is to provide classes of continuous spherical and elliptical distributions with explicit formulas for marginal densities of arbitrary dimension. We are interested in consistent families of generators for random vectors X ∼ SC p ( g p ) , where every marginal distribution of X of dimension m is distributed as SC m ( h ) and h also belongs to the consistent family. We search for an algorithm by which, starting from g 1 , a family { g p } p ≥ 1 of density generators can be derived. Here, ideas of Kano’s [8] paper are utilized. In Section 2, we study the properties of consistent families of generator functions. Section 3 is devoted to the construction of families. The random number generation of spherically distributed vectors is briefly discussed in Section 4. A survey of consistent families of elliptical distributions with explicit formulas can be found in Sections 5 and 6. Section 7 presents the proofs of the presented theorems.

2 Characterization of consistent generator families

In this section, we consider a family of non-negative functions { g p } p ∈ I , where I = { 1 , 2 , … , p 0 } , p 0 ≥ 3 , or I = { 1 , 2 , … } = N , i.e., p 0 = ∞ . We call a family { g p } p ∈ I of density generators consistent if

for all z 1 , … , z p ∈ R , 1 ≤ p < p 0 . The notion of consistency goes back to Kano [8]. Now we give the most important statements of Kano’s paper.

Equation (2.3) can be interpreted in a manner that allows g p to be represented as a Laplace-Stieltjes transform:

where F ˜ p ( t ) = ∫ 0 t ( s ∕ π ) p ∕ 2 d F 0 ( 2 s ) . Unfortunately, in many cases, F 0 and F ˜ p are not of explicit form. If F 0 has a density, then it is denoted by f 0 . In this case, the function t ↦ ℒ − 1 ( g p ) ( t ) = 2 t π p ∕ 2 f 0 ( 2 t ) is the inverse Laplace transform of g p , which implies

The following Proposition 2.2 provides a formula for the marginal densities of spherical distributions. This statement follows directly from (2.1).

3 Construction of consistent density generators

Consistent families are connected with completely monotonic functions on [ 0 , ∞ ) (cf. [11]):

DEFINITION: A function f : [ 0 , ∞ ) → [ 0 , ∞ ) is completely monotonic if all derivatives exists on [ 0 , ∞ ) and

The Bernstein-Widder theorem (cf. [11], Theorem A) gives an important assertion about the characterization of completely monotonic functions:

The central idea in this section is to determine the function g 1 and to provide evaluation formulas for the other generators { g p } p ≥ 2 to obtain a consistent family. The following Theorem 3.2 provides an equivalent characterization of families of functions satisfying the consistency condition (2.2) in the case of an infinite family { g p } with p 0 = ∞ .

Furthermore, an equation describing the relation between g p and g p + 1 can be established using fractional derivatives:

Considering the construction of elliptical densities, functions g p have to satisfy the additional condition (1.1). Theorem 3.4 contains the corresponding statement.

On the basis of Theorem 3.4, we can provide algorithms for the construction of functions g p .

Algorithm for infinite p 0 ̲

1. Choose a completely monotonic function g 1 with g 1 ( 0 + ) < + ∞ , (3.3) and lim r → ∞ g 1 ( r ) = 0 .

2. Evaluate g 3 ( r ) = − 1 π g 1 ′ ( r ) .

3. Evaluate g 2 ( r ) = 2 ∫ 0 ∞ g 3 ( r + y 2 ) d y .

4. For k = 1 , 2 , … , do.

   g 2 k + 2 ( r ) = − 1 π g 2 k ′ ( r ) , g 2 k + 3 ( r ) = − 1 π g 2 k + 1 ′ ( r ) .

1. Choose a non-negative differentiable function g 1 with g 1 ( 0 + ) < + ∞ , (3.3) and lim r → ∞ g 1 ( r ) = 0 .

2. Evaluate g 3 ( r ) = − 1 π g 1 ′ ( r ) and check g 3 ( r ) ≥ 0 for r ≥ 0 .

3. Evaluate g 2 ( r ) = 2 ∫ 0 ∞ g 3 ( r + y 2 ) d y and check g 2 ( r ) ≥ 0 for r ≥ 0 .

4. For k = 1 , 2 , … , ⌊ ( p 0 − 3 ) ∕ 2 ⌋ , do

   g 2 k + 2 ( r ) = − 1 π g 2 k ′ ( r ) , g 2 k + 3 ( r ) = − 1 π g 2 k + 1 ′ ( r ) ,

   and check g 2 k + 2 ( r ) , g 2 k + 3 ( r ) ≥ 0 for r ≥ 0 .

At first glance, it seems that for finite p 0 , more checks are required in the algorithms than in the infinite case. Here, it should be noticed that completely monotonicity of g 1 has a lot of consequences under (3.1) and (3.2): g p is completely monotonic and g p ( r ) ≥ 0 in view of Lemma 7.2, g p ( 0 ) = g p ( 0 + ) < + ∞ , and g p ( r ) ≥ 0 for r ≥ 0 . The remarks on page 2 of the study by Miller and Samko [14] show that completely monotonic functions on [ 0 , ∞ ) cannot vanish on an interval and are strictly positive on [ 0 , ∞ ) . Therefore, in the case p 0 = ∞ , the density generators from the aforementioned algorithm are strictly positive on [ 0 , ∞ ) .

Now we consider the problem of how to obtain other consistent families from two given families. One approach is to deal with mixtures of scale transformed generators.

In this proposition, a special case is given by h p ( r ) = a p ∕ 2 g p ( a r ) , where a > 0 is a parameter. The proof can be done in a straightforward way.

4 Random number generation

Theorems 1.1 and 1.2 can be used for the generation of spherical or elliptical distributions. Now we show how to generate X ∼ SC d ( g d ) .

Algorithm ̲

1. Generate V ∼ N ( 0 , I ) deploying standard algorithms, e.g., the Box-Muller method

2. U = 1 ∥ V ∥ V

3. Generate R having density according to (1.4)

4. X = R U

5 Models for consistent generator functions on the infinite interval [ 0 , ∞ )

In the following, we do not consider scale parameters for the generator families because they are taken into account in the definition of elliptical densities. Moreover, the normalizing constants are chosen such that (3.3) is satisfied. All computations were done by the computer algebra system (CAS) Mathematica. Let R p denote the random radius of a p -dimensional spherical random vector. The Appendix contains figures of a couple of the generator functions of this section.