Tail Conditional Expectation and Tail Variance for Extended Generalized Skew-Elliptical Distributions

This study derives explicit expressions for the Tail Conditional Expectation (TCE) and Tail Variance (TV) within the framework of the extended generalized skew-elliptical (EGSE) distribution. The EGSE family generalizes the class of elliptical distributions by incorporating a selection method, thereby allowing simultaneous and flexible control over location, scale, skewness, and tail heaviness in a unified parametric setting. As notable special cases, our results encompass the extended skew-normal, extended skew-Student-t, extended skew-logistic, and extended skew-Laplace distributions. The derived formulas extend existing results for generalized skew-elliptical distributions and reduce, to a considerable extent, the reliance on numerical integration, thus enhancing their tractability for actuarial and financial risk assessment. The practical utility of the proposed framework is further illustrated through an empirical analysis based on real stock market data, highlighting its effectiveness in quantifying and contrasting the heterogeneous tail risk profiles of financial assets.