Estimating a distribution function for discrete data subject to random truncation with an application to structured finance

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.