The deficiency of sample quantiles with respect to quasiquantiles is investigated under the assumption that the true density function has bounded derivatives. Then the sample quantile is still an efficient estimator of the true quantile but the relative deficiency of sample quantiles with respect to suitably defined quasiquantiles quickly tends to infinity for increasing sample sizes. If the second derivative of the true density function is bounded, then adaptive estimators will be found which are of a better performance than quasiquantiles. Corresponding results are derived for two-sided confidence intervals which are based on quasiquantiles and adaptive estimators.
"Estimation of Quantiles in Certain Nonparametric Models." Ann. Statist. 8 (1) 87 - 105, January, 1980. https://doi.org/10.1214/aos/1176344893