chapter  3
22 Pages

Skew-Elliptical Distributions

It is well-known that symmetric distributions are not suitable for modeling all types of data. Therefore, there is a strong motivation to construct skewed distribution with nice properties. Azzalini (1985, 1986) formalized the univariate skew-normal distribution having the probability density function (pdf) of the form 2φ(x)Φ(αx), x, α ∈ R. Azzalini and Dalla Valle (1996) extended the results to the multivariate setting with the pdf of the form 2φp(x;µ,Ω)Φ(αT (x − µ)), x,α ∈ Rp, which has been generalized later by many authors. To incorporate departures from normality, the first approach taken is to consider elliptical distributions. Consider a p-dimensional random vector x having a pdf of the form

f(x|µ,Ω, g(p)) = |Ω|− 12 g(p)((x− µ)TΩ−1(x− µ)), x ∈ Rp, (3.1) where g(p)(u) is a non-increasing function from R+ to R+ defined by

g(p)(u) = Γ(p/2) πp/2

g(u; p)∫∞ 0

rp/2−1g(r; p)dr , (3.2)

and g(u; p) is a non-increasing function from R+ to R+ such that the integral

rp/2−1g(r; p)dr exists. We will always assume the existence of the pdf f(x|µ,Ω, g(p)). The function g(p) is often called the density generator of the random vector x. Note that the function g(u; p) provides the kernel of x and other terms in g(p) constitute the normalizing constant for the pdf f . In addition, the function g, hence g(p), may depend on other parameters that would be clear from the context. For example, in the case of the t distribution, the additional parameter will be the degrees of freedom. The density f defined above represents a broad class of distributions called the elliptically contoured distributions, and we will use the notation x ∼ ECp(µ,Ω, g(p)) from now on in this chapter. Let F

( x|µ,Ω, g(p)) denote the cumulative

distribution function (cdf) of x. In the fields of biology, economics, psychology, sociology, and so on, of-

ten the application backgrounds exhibit non-normal error structures. The skewness is one of the significant aspects for such departures from normality. It is suggested that the error structures in this case should be handled

ate normality framework. Arnold and Beaver (2002) gave a comprehensive review of the literature on multivariate skewness construction, interpretation, and property research. Their explanation for the skewness mechanism is related to hidden truncation and/or selective reporting. A very convenient, flexible, and well-behaved multivariate distribution class is the multivariate skew-elliptical distribution (see Branco and Dey, 2001) which can be used for modeling skewed distributions with heavy tails. Branco and Dey (2001) presented a class of multivariate skew-elliptical distributions that includes several unimodal elliptical and spherical distributions. Their method is developed by introducing a skewness parameter. The new distribution brings additional flexibility of modeling skewed data that can be used in regression and calibration in the presence of skewness. They studied the properties of such skew-elliptical distributions, and we discuss some of them in Section 3.4. Arnold and Beaver (2002) interpreted univariate skew-normal, non-normal univariate skewed distributions, multivariate skew-normal, and non-normal skewed multivariate distributions. They also discussed their construction of skew-elliptical distributions, which included the method of Branco and Dey (2001) as a special case.