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Proper econometric analysis should be informed by data structure. Many formsof financial data are recorded in discrete-time and relate to products of afinite term. If the data comes from a financial trust, it will often be furthersubject to random left-truncation. While the literature for estimating adistribution function from left-truncated data is extensive, a thoroughliterature search reveals that the case of discrete data over a finite numberof possible values has received little attention. A precise discrete frameworkand suitable sampling procedure for the Woodroofe-type estimator for discretedata over a finite number of possible values is therefore established.Subsequently, the resulting vector of hazard rate estimators is proved to beasymptotically normal with independent components. Asymptotic normality of thesurvival function estimator is then established. Sister results for theleft-truncating random variable are also proved. Taken together, the resultingjoint vector of hazard rate estimates for the lifetime and left-truncationrandom variables is proved to be the maximum likelihood estimate of theparameters of the conditional joint lifetime and left-truncation distributiongiven the lifetime has not been left-truncated. A hypothesis test for the shapeof the distribution function based on our asymptotic results is derived. Such atest is useful to formally assess the plausibility of the stationarityassumption in length-biased sampling. The finite sample performance of theestimators is investigated in a simulation study. Applicability of thetheoretical results in an econometric setting is demonstrated with a subset ofdata from the Mercedes-Benz 2017-A securitized bond.

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