Estimators and Their Asymptotic Properties for Quantile Difference with Left Truncated and Right Censored Data

We investigate the asymptotic properties of the estimators of quantile difference based on left truncated and right censored data. The TJW product-limit estimator of the distribution function with the left truncated and right censored data is used to provide the empirical estimator of the quantile difference. Meanwhile, another smoothed kernel estimator for the quantile difference is established. Using the theory of empirical process, the expressions of the asymptotic bias and variance of the two estimators are derived. The large sample properties, such as consistency and asymptotic normality, for the estimators are obtained. A small simulation study shows that in the sense of mean squared loss, the smoothed estimator is more efficient than the non-smoothed estimator.