Nonparametric estimation of quantile density function for truncated and censored data

In this paper we investigate the asymptotic properties of two types of kernel estimators for the quantile density function when the data are both randomly censored and truncated. We derive some laws of the logarithm for the maximal deviation between fixed bandwidth kernel estimators or random bandwidth kernel estimators and the true underlying quantile density function. Extensions to higher derivatives are included. The results are used to obtain the optimal bandwidth with respect to almost sure uniform convergence.