The Convergence Rate and Asymptotic Distribu- Tion of Bootstrap Quantile Variance Estimator for Importance Sampling

Importance sampling is a widely used variance reduction technique to compute sample quantiles such as value-at-risk. The variance of the weight sample quantile estimator is usually a difficult quantity to compute. In this paper, we present the exact convergence rate and asymptotic distributions of the bootstrap variance estimators for quantiles of weighted empirical distributions. Under regularity conditions, we show that the bootstrap variance estimator is asymptotically normal and has relative standard deviation of order O(n−1/4).