Quantile process for left truncated and right censored data

摘要

In this paper, we consider the product-limit quantile estimator of an unknown quantile function when the data are subject to random left truncation and right censorship. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate $O(frac{{(log n)^{3/2} }}{{n^{1/8} }})$ . A functional law of the iterated logarithm for the maximal deviation of the estimator from the estimand is derived from the construction.