By Kusuoka's representation theorem, the class of distortion risk measures with convex distortions coincides with the set of coherent risk measures that are law invariant and comonotonically additive. The class includes the renowned expected shortfall which has many nice features and is of frequent use in practice. To implement the risk management/regulatory procedure using risk measures, it is necessary to estimate the values of such risk measures. For a distortion risk measure, its form suggests a natural estimator which is a simple form of $L$-statistics. We have seen in our previous work that it has nice asymptotic properties with i.i.d.\ data. After reviewing these results briefly, we investigate the large sample properties of the estimator based on dependent data, especially GARCH sequences, which are often used for modelling financial time series data. Related issues such as semiparametric estimation with the extreme value theory and backtesting are briefly addressed.