The transformed Gram Charlier distribution: Parametric properties and financial risk applications

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