Inference with bivariate truncated data

In this paper we build on previous work for estimation of the bivariate distribution of the time variables T1 and T2 when they are observable only on the condition that one of the time variables, say T1, is greater than (left-truncation) or less than (right truncation) some observed time variable C1. In this paper, we introduce several results based on the Influence Curve (which we derive in this paper) of the NPMLE of the distribution F of (T1, T2) developed by van der Laan (van der Laan, 1996). Specifically we will: prove that the NPMLE is asymptotically equivalent to an estimator developed by Gürler (Gürler, 1997), derive the asymptotic distribution of the NPMLE based on its Influence Curve, present tests to determine the amount of dependence between T1 and T2, present the results of simulation studies that compare the NPMLE and Gürler's estimator and evaluate the performance of both the above mentioned tests and confidence intervals of F based on the asymptotic distribution of the NPMLE, and finally we will apply the methods in a data analysis in which we also point out practical issues that arise in the implementation of the estimator.