Tail densities of skew-elliptical distributions

Abstract

Skew-elliptical distributions constitute a large class of multivariate distributions that account for both skewness and a variety of tail properties. This class has simpler representations in terms of densities rather than cumulative distribution functions, and the tail density approach has previously been developed to study tail properties when multivariate densities have more tractable forms. The special skew-elliptical structure allows for derivations of specific forms for the tail densities for those skew-elliptical copulas that admit probability density functions, under heavy and light tail conditions on density generators. The tail densities of skew-elliptical copulas are explicit and depend only on tail properties of the underlying density generator and conditions on the skewness parameters. In the heavy tail case skewness parameters affect tail densities of the skew-elliptical copulas more profoundly than that in the light tail case, whereas in the latter case the tail densities of skew-elliptical copulas are only proportional to the tail densities of symmetrical elliptical copulas. Various examples, including tail densities of skew-normal and skew-t distributions, are given.

Statistical compact model extraction for skew‐normal distributions

“…Consider the input to the hidden unit (S j ) whose PDF f S j (x) is given by (21). The output (K j ) of the hidden unit is given by the non-linear transformation through the activation function ϕ(x) and its mean is given by…”

“…On the application side, GSE distributions have been used to model data in various scenarios like in risk theory [19] and Bayesian regression models [20] to name a few. GSE distributions allow flexibility in modelling data with asymmetry and heavy tails [21]. From the wide variety of distributions that belong to the GSE class we choose the SN distribution for the SCME problem at hand for the following reasons: (i) Historically, hardware measurements have shown that the variations in process parameters have been Gaussian in nature.…”

“…The proposed SCME algorithm should assign when there is no skew in the underlying process parameters, which is demonstrated in Section 6.2.1. (ii) We focus on modelling asymmetry in the distribution rather than the tail. Skew‐ t distributions could be used to model tail densities [21] but that would require a prohibitively large number of measurements from the manufacturing line to estimate the parameters of the distribution, which is not practical. However, skew‐ t densities could be considered in simulation‐based tail modelling [5] problems, which is not the focus of our work. (iii) Section 3.1 discusses the mathematics behind propagating mean of the chosen distribution through an ANN.…”

“…(ii) We focus on modelling asymmetry in the distribution rather than the tail. Skew‐ t distributions could be used to model tail densities [21] but that would require a prohibitively large number of measurements from the manufacturing line to estimate the parameters of the distribution, which is not practical. However, skew‐ t densities could be considered in simulation‐based tail modelling [5] problems, which is not the focus of our work.…”

Multivariate Skew t-Distribution: Asymptotics for Parameter Estimators and Extension to Skew t-Copula

“…Another useful property that makes multivariate t-distribution and skew t-distribution attractive in financial and actuarial applications is the possibility to take tail dependence between marginals into account. Tail dependence of multivariate tand skew t-distributions has been intensively studied in recent years, see Fung and Seneta [15] or Joe and Li [16], for example. In Kollo et al [17] it is shown that the upper tail dependence coefficient of the skew t-distribution can be several times bigger than the corresponding value of multivariate t-distribution.…”