Before revision:  In both cases, we show, under reasonable assumptions on the density function of X, how to compute a constant C such that |\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]| < C \epsilon^2, provided }\operatorname{rd}[ |X^k-\operatorname{rd}(X)^{k}| ] are also given} .
After revision:  In both cases, we show, under reasonable assumptions on the density function of X, how to compute a constant C such that |\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]| < C \epsilon^2, provided }\operatorname{rd}(x) - x| \leq \epsilon E(x). Asymptotic and non-asymptotic bounds for the absolute moments \mathbb{E[ |X^k-\operatorname{rd}(X)^{k}| ] are also given} .