Estimating the variance of bootstrapped risk measures

Nonparametric variance estimation for the distortion risk measure can be readily done through the bootstrap or the nonparametric delta method based on the influence function. The same task for the bootstrapped risk measures, however, has been relatively unexplored in the literature. In this paper we analytically derive the influence function of the exactly bootstrapped quantile and later extend this to the L-estimator class. The resulting formula provides an alternative method to estimate the variance of the bootstrapped risk measures, or the whole L-estimator class in an analytic form. A simulation study shows that this new method is comparable to the ordinary resampling-based bootstrap method.