Plug-in bandwidth selection rules for the kernel quantile estimator

The large sample theory of the kernel quantile estimator is extended by separately treating the cases where the underlying density has critical points. Our attempts of improving the quality of quantile estimation resulted in proposing the beta distribution's bandwidth selection method that is quite successful in the case of a normal distribution. The performance of the beta reference rule is compared to that of two other plug-in type bandwidth selectors. Based on the theoretical and numerical results, we provide certain recommendations regarding using different type of estimators in practice. In particular, the beta reference rule and truncated normal rule are recommended for estimating the quantiles of the distributions close to normal. The sample quantile estimator worked well in all simulation settings and is thus recommended as a default quantile estimation method in a non-normal case.