Unified estimators of smooth quantile and quantile density functions

Diverse smooth estimators of quantile functions are unified into generalized kernel smoothers of the sample quantile function. Sup-norm convergence of the quantile function estimators and the derived quantile density function estimators are established. The convergence rates reveal the simultaneous uniform in-probability consistency of both estimators on any compact set inside the open unit interval. A Berry-Esseen-type theorem of the quantile function estimators is established; the asymptotic deficiency of sample quantiles relative to the smooth estimators and optimal smoothing rate are determined. Oscillation behavior of the estimators in finite samples (monotonicity, convexity, etc.) is investigated, and is related to certain algebraic properties of the smoothing kernel.