In this paper we study an extension of the Gram–Charlier (GC) density in Jondeau and Rockinger (2001) which consists of a Gallant and Nychka (1987) transformation to ensure positivity without parameter restrictions. We derive its parametric properties such as unimodality, cumulative distribution, higher-order moments, truncated moments, and the closed-form expressions for the expected shortfall (ES) and lower partial moments. We obtain the analytic $k$th order stationarity conditions for the unconditional moments of the TGARCH model under the transformed GC (TGC) density. In an empirical application to asset return series, we estimate the tail index; backtest the density, VaR and ES; and implement a comparative analysis based on Hansen’s skewed-t distribution. Finally, we present extensions to time-varying conditional skewness and kurtosis, and a new class of mixture densities based on this TGC distribution.

在本文中，我们研究了 Jondeau 和 Rockinger (2001) 中 Gram-Charlier (GC) 密度的扩展，它由 Gallant 和 Nychka (1987) 转换组成，以确保没有参数限制的积极性。我们推导出其参数属性，例如单峰性、累积分布、高阶矩、截断矩以及预期短缺 (ES) 和较低部分矩的封闭形式表达式。我们得到解析$克$TGARCH 模型在转换后的 GC (TGC) 密度下无条件矩的 th 阶平稳条件。在资产收益序列的实证应用中，我们估计尾部指数；回测密度、VaR 和 ES；并基于 Hansen 的偏 t 分布进行比较分析。最后，我们提出了对时变条件偏度和峰度的扩展，以及基于此 TGC 分布的一类新的混合密度。