The cumulative quantile regression function with censored and truncated response

In this paper, we propose a nonparametric estimator of the cumulative quantile regression (CQR) function when the response is subjected to random truncation and censorship. The observed responses are weighted by the increments of the product-limit estimator for the underlying response distribution. Strong Gaussian approximations of the associated weighted partial sum process and the CQR process are established under appropriate assumptions. A functional law of the iterated logarithm for the CQR process is also derived. The construct provides a foundation for the asymptotic theory of functional statistics based on these processes. © Allerton Press, Inc., 2015.