Materials for “ Improved Doubly Robust Estimation when Data are Monotonely Coarsened , with Application to Longitudinal Studies with Dropout ”

We provide an argument that (6) of the main paper is an unbiased estiamting function for ξ when the coarsening probabilities may not be correctly specified, so that the posited probabilities π{r, Gr(Z)} 6 = π0{r, Gr(Z)} for some r = 1, . . . ,M,∞, but the models hr{Gz(Z), ξ} are correct, and that estimators for ξ based on (6) will converge in probability to ξ0 for arbitrary choice of the functions qr {Gr(Z), ξ}. The argument proceeds by a series of iterated conditional expectations to show that (6) has mean zero under these conditions. When r = M , the summand in (6) is