A unified view on skewed distributions arising from selections

Abstract

Parametric families of multivariate nonnormal distributions have received considerable attention in the past few decades. The authors propose a new definition of a selection distribution that encompasses many existing families of multivariate skewed distributions. Their work is motivated by examples that involve various forms of selection mechanisms and lead to skewed distributions. They give the main properties of selection distributions and show how various families of multivariate skewed distributions, such as the skew‐normal and skew‐elliptical distributions, arise as special cases. The authors further introduce several methods of constructing selection distributions based on linear and nonlinear selection mechanisms.

Scaled Largest Eigenvalue Detection for Stationary Time-Series

“…equation (10)). However, as explained in [26], the consequences of this singularity are only of computational nature and the distribution F λ1 (x) still follows a SLCT-N distribution. As a result, our model can be used to approximate the distribution of the SLE detector when the received signal is first modified by the spatial sign function, which we will use to analyze the performance of the modified detector in the presence of impulsive noise.…”

“…We introduce here the selection normal distribution (also called SUN distribution), as presented in [26], [29].…”

A non-Gaussian multivariate distribution with all lower-dimensional Gaussians and related families

“…In the following sections, we study the symmetry of this distribution (Sections 3 and 4), its moments (Section 5) and other properties related to its probability distribution function (Section 6). A direction of extending normal distributions to other symmetric cases is by considering elliptically contoured distributions (Fang, Kotz and Ng, 1990) and asymmetric distributions via selections (Arellano-Valle, Branco and Genton, 2006). In particular, we note that the multivariate skew-normal distribution (Azzalini and Dalla Valle, 1996) and other multivariate skewed distributions are special cases of these selections (Section 7).…”

A general setting for symmetric distributions and their relationship to general distributions

“…For any G of type (3), there are infinitely many G 0 and w such that G(x) = G 0 {w(x)}. If G(x) = 1/2 or, equivalently w(x) = 0, then f = f 0 .…”

“…A corollary of construction (4) is the modulation-invariance property, which states that t(Z 0 ) and t(Z) have the same distribution for any R q -valued function t such that t(x) = t(−x) for all x in R d . Even more general formulations of (2) exist; see [3]. However, almost all formulations assume central symmetry of f 0 , i.e., f 0 (x) = f 0 (−x).…”