Inference with Bivariate Truncated Data

Inthis paper we build on previous work for estimation of the bivariatedistribution of the time variables T ₁ and T ₂when they are observable only on the condition that one of thetime variables, say T ₁, is greater than (left-truncation)or less than (right truncation) some observed time variable C ₁.In this paper, we introduce several results based on the InfluenceCurve (which we derive in this paper) of the NPMLE of the distributionF of (T ₁,T ₂) developed by van derLaan (van der Laan, 1996). Specifically we will: prove that theNPMLE is asymptotically equivalent to an estimator developedby Gürler (Gürler, 1997), derive the asymptotic distributionof the NPMLE based on its Influence Curve, present tests to determinethe amount of dependence between T ₁ and T ₂,present the results of simulation studies that compare the NPMLEand Gürler's estimator and evaluate the performance of boththe above mentioned tests and confidence intervals of Fbased on the asymptotic distribution of the NPMLE, and finallywe will apply the methods in a data analysis in which we alsopoint out practical issues that arise in the implementation ofthe estimator.