A bootstrap-based bandwidth selection rule for kernel quantile estimators

The quantile has been widely used to quantify the uncertainty in many fields. In this paper, we study the estimation of quantiles via kernels, especially for extreme quantiles, and propose a bootstrap-based bandwidth selection (BBS) method for it. This method employs bootstrap sampling of data and least-squares regression to estimate the unknown bandwidth parameter in the kernel, which plays a crucial role in kernel smoothing. From a theoretical perspective, we establish a data-driven and bootstrap-based kernel quantile estimator and provide its asymptotic bias and variance, based on which the proposed method is shown to lead to the asymptotically optimal bandwidth selection in terms of minimizing the mean squared error. Numerical experiments demonstrate that the BBS method works well in both bandwidth selection and extreme quantile estimation.