Multivariate extremes of generalized skew-normal distributions

Abstract

We explore extremal properties of a family of skewed distributions extended from the multivariate normal distribution by introducing a skewing function π. We give sufficient conditions on the skewing function for the pairwise asymptotic independence to hold. We apply our results to a special case of the bivariate skew-normal distribution and finally support our conclusions by a simulation study which indicates that the rate of convergence is quite slow.

Multivariate extreme models based on underlying skew-t and skew-normal distributions

“…When a skew-normal random vector is considered, the resulting marginal maxima are still asymptotically independent, as has been shown by Lysenko et al [31] and Bortot [7]. Therefore some modifications, as provided by Hüsler and Reiss [23], are required in order to obtain a non-trivial limit.…”

“…While the extremal behaviour of multivariate symmetric models such as the normal and t distributions has been extensively studied, their skewed versions have not been widely characterized. Attention to these skewed versions has only recently been paid by Chang and Genton [9], Lysenko et al [31] and Bortor [7]. The absence of the extremal versions of these models has motivated our interest in these families of distributions.…”

“…A number of different applications are well-documented by Azzalini and Capitanio [2] and Genton [18], whose work has justified the usefulness of this family of models. In the bivariate case, [31,7] show that, while the SN distribution captures dependence for a wider range of ''extreme'' observations than does the Gaussian distribution, it fails to describe the tail dependence among extreme values. Instead, the skew-t (ST ) distribution, that is, the asymmetric version of the t [3,1] turns out to have heavy tails similar to the generating family, therefore the tail dependence can be appropriately assessed.…”

Extremal properties of the multivariate extended skew-normal distribution, Part B

“…(2) is equal to the product of its marginal distributions (e.g Lysenko et al, 2009, Beirlant et al, 2004. However, Beranger et al (2017) showed that for the skew-normal case, the rate of convergence to zero of the upper-tail dependence function χ(u) in (4) depends on the slant parameters α, and depending on the sign of the elements of α, this can occur at a faster or slower rate than that of the normal case.…”

Asymptotic independence for unimodal densities

“…, ξ m are linearly independent. Asymptotic independence of the skew-normal distribution has been partially proven in [17] using a direct analytic approach based on Sibuya's condition. Example 5.4.…”