The Distribution of Quadratic Expressions in Elliptically Contoured Vectors

A general representation of quadratic expressions in possibly singular elliptically contoured random vectors, as well as a procedure for the numerical evaluation of their distributions, are proposed in this paper. First, such quadratic expressions are represented as the difference of two positive definite elliptically contoured quadratic forms plus an independently distributed linear combination of spherically distributed random variables. Their distributions are then determined from a representation of elliptically contoured vectors in terms of scale mixtures of Gaussian vectors. Quadratic forms and quadratic expressions in various types of elliptically contoured vectors are considered. An accurate moment-based approximation to their density function is also provided. Several numerical examples illustrate the results.


Introduction
Several fields of application involve elliptically contoured distributions, including, for instance, anomalous change detection in hyperspectral imagery: Theiler et al. (2010); option pricing: Hamada and Valdez (2008); filtering and stochastic control: Chu (1973); random input signal: McGraw and Wagner (1968); financial analysis: Zellner (1976) and the references therein; the analysis of stock market data: Mandelbrot (1963) and Fama (1965); and Bayesian Kalman filtering: Girón and Rojano (1994).Additionally, studies on the robustness of statistical procedures when the probability model departs from the multivariate normal distribution to the broader class of elliptically contoured distributions were carried out by King (1980) and Osiewalski and Steel (1993).Several multivariate applications are also discussed in Devlin et al. (1976).Results related to regression analysis can be found, for example, in Fraser and Ng (1980).Heavy-tailed time series models were discussed in Resnick (1997).A new family of life distributions, generated from an elliptically contoured distribution, is discussed by Díaz-García and Leiva-Sánchez (2005).Recently, Ip et al. (2007) derived some results applicable to Bayesian inference for a general multivariate linear regression model with matrix variate elliptically distributed errors.In fact, the class of elliptically contoured distributions, which contains the multivariate normal distribution, enjoys several of its properties while allowing for more flexibility in modeling various random processes.
A p-dimensional vector X has an elliptically contoured or elliptical distribution with mean vector μ and scale parameter matrix Σ if its characteristic function φ(t) can be written as where μ is a p-dimensional real vector, Σ is a p × p nonnegative definite matrix and ξ(•) is a nonnegative function, see, for instance, Cambanis et al. (1981); this will be denoted X ∼ C p (μ, Σ; ξ).
Moreover, the densities associated with p-dimensional elliptically contoured vectors X are of the form h((x − μ) Σ −1 (x − μ)) where h(•) is a density defined on (0, ∞) whose (p/2 − 1) th moment exists, see for example Fang et al. (1990), Section 2.2.3.In particular, when μ is the null vector and Σ is the identity matrix of order p, X is said to have a spherically symmetric or spherical distribution; this will be denoted X ∼ S p (ξ).
A decomposition of quadratic expressions in possibly singular elliptically contoured vectors is introduced in Section 2 and representations of functions of elliptically contoured vectors such as the moments of a quadratic form, are obtained in Section 3. A density approximation methodology that combines these results is described and illustrated by several numerical examples in Section 4.
The distributional results derived in this paper for quadratic forms in elliptically contoured random vectors not only extend, but also make use of, their Gaussian counterparts.Given that elliptically contoured distributions are utilized as models in a host of applications, and quadratic forms are ubiquitous in statistics, the result presented herein should prove useful in a variety of contexts and lead to the development of improved statistical inference techniques.
As a particular case, when α = 0, a = 0 and d = 0, one has the following decomposition for the quadratic f orm X AX in the possibly singular elliptically contoured vector X ∼ C p (μ, Σ; ξ), Σ being of rank r ≤ p: where , θ is the number of null eigenvalues of AΣ, the λ j 's and b * j 's being as previously defined,

Elliptically Contoured Distributions as Scale Mixtures of Gaussian Vectors
Normal scale mixtures have the stochastic representation μ + Σ 1/2 L Z, where μ is the mean of the distribution, Σ 1/2 is such that Σ 1/2 (Σ 1/2 ) = Σ, the positive semidefinite scale parameter matrix of the distribution, Z is a standard Gaussian random vector, and L is a positive random variable that is distributed independently of Z.When Σ is positive definite, the density function of Y ∼ C p (μ, Σ; ξ) can be expressed in terms of a scale mixture of normal densities as follows: where U(•), the distribution function of L 2 , is such that U(0) = 0.This representation can be found, for example, in Muirhead (1982).A result due to Chu (1973) is extended to non-central elliptically contoured distributions in the next theorem.This theorem enables one to express various distributional results involving elliptically contoured vectors in terms of their Gaussian counterparts.
Theorem 1 Let Y ∼ C p (μ, Σ; ξ), with Σ > 0, h(y) denote the density of y and f (s) be h(y) wherein (y − μ) Σ −1 (y − μ)/2 is replaced by s.Then, when the inverse Laplace transform of f (s) exists, the density of Y denoted by h(y) has the following integral representation: where η Y (μ, t −1 Σ) denotes the density function of a p-dimensional Gaussian random vector with mean μ and covariance matrix t −1 Σ, and the weighting function w(t) is obtained as L −1 ( f (s)) representing the inverse Laplace transform of f (s).
In fact, L −1 ( f (s)) exists whenever f (s) is an analytic function and f (s) is O(s −k ) as s → ∞ for k > 1; for additional properties of the Laplace transform and its inverse, one may refer to Gradshteyn and Ryzhik (1980), Chapter 17.It follows from Theorem 1 that an elliptical distribution is completely specified by its mean μ, scale parameter matrix Σ and its weighting function w(t), whenever the latter exists.On letting t = 1/r and defining w(t) to be the density function of 1/L 2 , it is seen that ( 3) and ( 4) are equivalent.On integrating h(y) as defined in Theorem 1 over R p and interchanging the order of integration, one can easily establish that w(t) integrates to 1. Thus, w(t) can be regarded as a weighting function.Explicit representations of w(t) are given in Table 2 for several p-dimensional elliptically contoured distributions.
Theorem 1 enables one to determine the distribution of functions of elliptically contoured vectors in terms of their Gaussian counterparts.For instance, let Y ∼ C p (μ, Σ; ξ) and its associated weighting function be w(t).Then, the moment-generating function of the non-central quadratic form Y AY can be obtained as follows: where Similarly, the moments of Y AY can be evaluated as follows: where W ∼ N p (μ, t −1 Σ) and E (W AW) h can be determined from (7).
In general, the moments of a random variable can be obtained from its cumulants by means of a recursive relationship derived in Smith (1995), which can also be deduced for instance from Theorem 3.2b.2 in Mathai and Provost (1992).For example, the h th moment of Q(W) = W AW is given by where

Illustrative Examples
Four numerical examples involving quadratic forms and quadratic expressions in various types of elliptically contoured vectors are presented in this section.The steps of the proposed methodology for determining their distributions are described in the first example.
Example 1 Consider the quadratic form Q I (X) = X AX where X has a noncentral t−distribution with 10 degrees of freedom whose density function is as given in Table 1 with The proposed methodology comprises the following steps: 2) The moments of Q I i (W i ), i = 1, 2 are determined from Equations ( 6) and ( 7). 3) A generalized gamma density function, is taken as base density for Q I i (W i ), i = 1, 2. 4) The parameters α, β and γ are determined by simultaneously solving the following nonlinear equations μ j = m j for j = 1, 2, 3, where are the moments associated with a generalized gamma density function and μ j can be determined from the recursive formula (7).
5) A polynomial adjustment of degree d can be made as explained in the Appendix, the resulting density approximation being in this case, we set d = 7. 6) Given the density approximations determined for Q I 1 (W 1 ) and Q I 2 (W 2 ), the approximate density of the difference is obtained by applying the transformation of variables technique.Shifting this density by κ then yields the desired approximation.
Certain values of the resulting approximate distribution function of Q I (X) are included in Table 1.The percentiles were obtained by simulation on the basis of 1,000,000 replications.The plot shown in Figure 1 confirms that the proposed approach yields a very accurate approximation to the distribution of Q I (X).Example 2 Consider the quadratic form Q II (X) = X AX where X is a contaminated normal random vector as specified in Table 2, for which φ = 0.4, μ = (1, 2, 3) , It this case, a gamma distribution (as defined by ( 8) with γ = 1) was utilized as base density to obtain an approximate distribution for each quadratic form in decomposition of Q II (X).Letting the integer moments of a non-negative definite quadratic form be denoted by μ j , j = 1, 2, . . ., a gamma approximation can be specified by equating its first two moments to μ 1 and μ 2 , respectively, and solving for α and β, that is, αβ = μ 1 and α(α + 1)β 2 = μ 2 , which yields The methodology described in Example 1 was applied in conjunction with polynomial adjustments of degree six to determine the approximate distribution of Q II (X).The plot shown in Figure 2 indicates that the resulting approximation is very accurate.
The Generalized Slash Distribution νs Example 3 Consider the quadratic form Q III (X) = X AX where X follows a generalized slash distribution whose density function is as defined in Table 2 By making use of the weighting function associated with the generalized slash distribution in order to determine the moments (Equation ( 7)) of the quadratic forms occurring in its decomposition and implementing the steps described in Example 1 in conjunction with a gamma distribution or a generalized gamma distribution whose associated densities are taken as base densities, one can determine an approximate distribution for Q III (X).
The left and right panels of Figure 3 respectively show the distribution functions resulting from gamma and generalized gamma approximations, which are superimposed on the simulated distribution function determined on the basis of 1,000,000 replications.When Σ p×p is a singular matrix of rank r < p, we make use of the spectral decomposition theorem to express Σ as UWU where W is a diagonal matrix whose first r diagonal elements are positive, the remaining diagonal elements being equal to zero.Next, we let B * p×p = UW 1/2 and remove the last p − r columns of B * , which are null vectors, to obtain the matrix B p×r .Then, it can be verified that Σ = BB .In this case, the matrices B and P were found to be

Figure 2 .
Figure 2. Simulated cdf of Q II (X) and cdf approximation (dots)

Figure 3 .
Figure 3. Simulated cdf of Q III (X) and approximations based on polynomially adjusted gamma (left panel) and generalized gamma (right panel) distributions (dots)

Table 1 .
Approximate cdf of Q I (X) evaluated at certain percentiles obtained by simulation (Simul%) Figure 1.Simulated cdf of Q I (X) and cdf approximation (dots)