There are several ways to establish the asymptotic normality of $L$-statistics, which depend on the choice of the weights-generating function and the cumulative distribution selection of the underlying model. In this study, we focus on stablishing computational formulas for the asymptotic variance of two robust $L$-estimators: the method of trimmed moments (MTM) and the method of winsorized moments (MWM). We demonstrate that two asymptotic approaches for MTM are equivalent for a specific choice of the weights-generating function. These findings enhance the applicability of these estimators across various underlying distributions, making them effective tools in diverse statistical scenarios. Such scenarios include actuarial contexts, such as payment-per-payment and payment-per-loss data scenarios, as well as in evaluating the asymptotic distributional properties of distortion risk measures. The effectiveness of our methodologies depends on the availability of the cumulative distribution function, ensuring broad usability in various statistical environments.