Geskus (2011, Biometrics, 67, 39-49) studied estimation of the Fine-Gray model for the cumulative incidence function with left truncated right censored competing risks data. The limiting distribution for an estimator base on weighting inversely using weights involving estimates of the joint distribution of the truncation and censoring times was derived via classical martingale theory with variance estimation based on martingale results. In this note, we demonstrate that martingale theory is not applicable and that other theoretical arguments, like those in Fine and Gray (1999), are needed to rigorously establish the asymptotic properties of the estimators and to construct valid variance estimators. For inverse probability of censoring weighted estimators, the common wisdom is that martingale theory fails because of estimation of the censoring distribution in the weights. For the Fine-Gray model, alternative theoretical developments are needed even with a known censoring distribution.
Li, Chenxi; Gray, Robert J.; and Fine, Jason P., "Reader Reaction: On Variance Estimation for the Fine-Gray Model" (January 2012). The University of North Carolina at Chapel Hill Department of Biostatistics Technical Report Series. Working Paper 29.