The literature for estimating a distribution function from truncated data is extensive, but it has given little attention to the case of discrete data over a finite number of possible values. We examine the Woodroofe-type estimator in this case and prove that the resulting vector of hazard rate estimators is asymptotically normal with independent components. Asymptotic normality of the survival function estimator is then established. Sister results for the truncation random variable are also proved. Further, a hypothesis test for the shape of the distribution function based on our results is presented. Such a test is useful to formally test the stationarity assumption in length-biased sampling. The finite sample performance of the estimators are investigated in a simulation study. We close with an application to an automotive lease securitization.

翻译：用于估计从缺勤数据中分配函数的文献十分广泛,但很少注意可能值数量有限的离散数据的情况。我们检查了此处的 Woodroofe 类型估计器,并证明由此产生的危险率估计器矢量与独立部件的正常程度不相上下。然后确定了生存函数估计器的正常性。还证明了短途随机变量的姐妹结果。此外,还提出了基于我们结果的分布函数形状的假设测试。这种测试有助于正式测试长度偏差抽样中的定点假设。在模拟研究中调查了估计器的有限样品性能。我们接近于汽车租赁证券化的应用。