An exact bootstrap approach towards modification of the Harrell-Davis quantile function estimator for censored data

A new kernel quantile estimator is proposed for right-censored data, which takes the form of Q̂(u; c) = Σ_j=1ⁿ T_(j)w_j(u,c), where w_j(u,c) is based on a beta kernel with bandwidth parameter c. The advantage of this estimator is that exact bootstrap methods may be employed to estimate the mean and variance of Q̂(u; c). It follows that a novel solution for finding the optimal bandwidth may be obtained through minimization of the exact bootstrap mean squared error (MSE) estimate of Q̂(u; c). We prove the large sample consistency of Q̂(u; c) for fixed values of c. A Monte Carlo simulation study shows that our estimator is significantly better than the product-limit quantile estimator Q̂_KM(u) = inf{t: F̂_n(t) ≥ u}, with respect to various MSE criteria. For general simplicity, setting c = 1 leads to an extension of classical Harrell-Davis estimator for censored data and performs well in simulations. The procedure is illustrated by an application to lung cancer survival data.