The Skew Normal multivariate risk measurement framework

Abstract

In this paper, we consider a random vector X = (X_1;X_2) following a multivariate Skew Normal distribution and we provide an explicit formula for the expected value of X conditioned to the event X <= X*, with X* in (-infty,+infty)^2. Such a conditional expectation has an intuitive interpretation in the context of risk measures.

Family of mean-mixtures of multivariate normal distributions: properties, inference and assessment of multivariate skewness

“…From Prékopa (1973) and An (1996), if the elements of a random vector are independent, and each has a log-concave density function, then their joint density is log-concave. We know that in the canonical form with PDF in (15), the random variables Z 1 , • • • , Z p are independent of each other. Log-concavity of MMNE distribution in the univariate case has been established in Proposition 3.1 of Negarestani et al (2019), and the PDF of the univariate normal distribution is also known to be log-concave.…”

“…The multivariate version of the skew-normal distribution has bean introduced by Azzalini and Dalla Valle (1996). This distribution has found diverse applications including portfolio optimization concepts and risk measurement indices in financial markets; see Bernardi et al (2020) and the references therein. A complete set of extensions of multivariate skew-normal distributions proposed in the last three decades can be found in Azzalini (2005) and Azzalini and Capitanio (2014).…”