Bahadur representation of the kernel quantile estimator under truncated and censored data

In this article the authors establish the Bahadur type representations for the kernel quantile estimator and the kernel estimator of the derivatives of the quantile function on the basis of left truncated and right censored data. Under suitable conditions, with probability one, the exact convergence rate of the remainder term in the representations is obtained. As a by-product, the LIL, the asymptotic normality for those kernel estimators are derived.