Before revision:  We derive asymptotic distributions for conditional trimmed mean and variance estimators, and use this to show asymptotic normality of the proposed teststatistic. Using financial empirical data , we compare our framework to various statistical tests including the Jarque-Bera test that is based on third and fourth moments .
After revision:  Next, we show how to use our framework to construct a powerful statistical normality test. In particular , we compare our framework to various statistical tests including the Jarque-Bera test that is based on third and fourth moments .