Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-$t$ distribution

Yet another skew-elliptical family but of a different kind: return to Lemma 1

In the context of modulated-symmetry distributions, there exist various forms of skew-elliptical families. We present yet another one, but with an unusual feature: the modulation factor of the baseline elliptical density is represented by a distribution function with an argument which is not an odd function, as it occurs instead with the overwhelming majority of similar formulations, not only with other skew-elliptical families. The proposal is obtained by going back to the use of Lemma~1 of Azzalini and Capitanio (1999), which can be seen as the general frame for a vast number of existing for…

On moments of folded and doubly truncated multivariate extended skew-normal distributions

This paper develops recurrence relations for integrals that relate the density of multivariate extended skew-normal (ESN) distribution, including the well-known skew-normal (SN) distribution introduced by Azzalini and Dalla-Valle (1996) and the popular multivariate normal distribution. These recursions offer a fast computation of arbitrary order product moments of the multivariate truncated extended skew-normal and multivariate folded extended skew-normal distributions with the product moments as a byproduct. In addition to the recurrence approach, we realized that any arbitrary moment of the …

多元统一偏斜椭圆分布的共轭性质

Conjugacy properties of multivariate unified skew-elliptical distributions

本文证明多元统一偏斜椭圆（SUE）分布在更广泛的多元回归与删失、二分化数据模型中具有共轭性，将贝叶斯共轭推断扩展至椭圆及偏斜椭圆族。

On the Computation of Multivariate Scenario Sets for the Skew-t and Generalized Hyperbolic Families

We examine the problem of computing multivariate scenarios sets for skewed distributions. Our interest is motivated by the potential use of such sets in the "stress testing" of insurance companies and banks whose solvency is dependent on changes in a set of financial "risk factors". We define multivariate scenario sets based on the notion of half-space depth (HD) and also introduce the notion of expectile depth (ED) where half-spaces are defined by expectiles rather than quantiles. We then use the HD and ED functions to define convex scenario sets that generalize the concepts of quantile and e…

椭圆分布及相关族的联合可混合性

Joint mixability of elliptical distributions and related families

本文提供椭圆分布联合可混合性的三种证明，推广至有限二阶矩分布，构造双峰对称分布解决开放问题，并研究slash-椭圆与偏椭圆分布及其多元扩展。

Optimal tests for elliptical symmetry: specified and unspecified location

Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most of the literature in the area indeed addresses the null hypothesis of elliptical symmetry with specified location and actually addresses location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam's as…

关于统一偏正态分布的非可辨识性

On the non-identifiability of unified skew-normal distributions

本文探讨了统一偏正态分布的非可辨识性，证明该问题在多种参数化及扩展分布中存在，并提出使模型可辨识的建议与可辨识子模型。

Generalized Location-Scale Mixtures of Elliptical Distributions: Definitions and Stochastic Comparisons

This paper proposes a unified class of generalized location-scale mixture of multivariate elliptical distributions and studies integral stochastic orderings of random vectors following such distributions. Given a random vector $\boldsymbol{Z}$, independent of $\boldsymbol{X}$ and $\boldsymbol{Y}$, the scale parameter of this class of distributions is mixed with a function $α(\boldsymbol{Z})$ and its skew parameter is mixed with another function $β(\boldsymbol{Z})$. Sufficient (and necessary) conditions are established for stochastically comparing different random vectors stemming from this cla…

贝塞尔函数表达的特征函数

The Bessel function expression of characteristic function

本文为椭圆及相关分布（含多元t、柯西等）的特征函数给出统一表达式，涉及贝塞尔型函数或广义超几何级数，并得到重要情形的闭式解。

Selection models under generalized symmetry settings

An active stream of literature has followed up the idea of skew-elliptical densities initiated by Azzalini and Capitanio (1999). Their original formulation was based on a general lemma which is however of broader applicability than usually perceived. This note examines new directions of its use, and illustrates them with the construction of some probability distributions falling outside the family of the so-called skew-symmetric densities.

On Rapid Variation of Multivariate Probability Densities

Multivariate rapid variation describes decay rates of joint light tails of a multivariate distribution. We impose a local uniformity condition to control decay variation of distribution tails along different directions, and using higher-order tail dependence of copulas, we prove that a rapidly varying multivariate density implies rapid variation of the joint distribution tails. As a corollary, rapid variation of skew-elliptical distributions is established under the assumption that the underlying density generators belong to the max-domain of attraction of the Gumbel distribution.