A Note On Self-Consistent Estimation of Censored Quantile Regression

Censored quantile regression has recently been studied under two major approaches: one is based on Portnoy (2003) that adopts self-consistent principle for handling censoring and the other one is based on Peng and Huang (2008) that utilizes the martingale structure of randomly censored data. Though numerical studies have suggested the close proximity between these two methods, their underlying connection has not yet been established. In this note, we propose to formulate self-consistent estimation of censored quantile regression through stochastic integral equations. This novel formulation not only renders a clear presentation of self-consistent estimation procedure but also aid in understanding the associated asymptotics, particularly uncovering the large sample equivalence between Portnoy’s approach and Peng-Huang’s approach. Furthermore, the proposed framework for censored regression quantiles is readily extended to other survival settings where the principle of self-consistency is applicable.