Estimation of the truncation probability with left-truncated and right-censored data

Abstract

Let (, , ) be i.i.d. random vectors such that (, ) is independent of and . Let F, Q, and G denote the common distribution function of , , and , respectively. For left-truncated and right-censored data, one can observe nothing if and observe ( ), with and , if . A problem of interest is the estimation of the truncation probability . Under the constraint that , Wang [Wang, M.-C., 1991, Nonparametric estimation from cross-sectional survival data. Journal of Americal Statistical Association, 86, 130–143.] suggested estimating α by α n = ∈ t [1−F n (s−)]dG n (s), where F n and G n are non-parametric maximum likelihood estimate of the distributions F and G, respectively. In this note, using inverse-probability-weighted estimators of F, G, and Q, we obtain two alternative representations, ˜α n and αˆ n , for α n . Simulation study shows that the estimator α n works satisfactorily for moderate sample size.