  Proper econometric analysis should be informed by data structure. Many forms
of financial data are recorded in discrete-time and relate to products of a
finite term. If the data comes from a financial trust, it will often be further
subject to random left-truncation. While the literature for estimating a
distribution function from left-truncated data is extensive, a thorough
literature search reveals that the case of discrete data over a finite number
of possible values has received little attention. A precise discrete framework
and suitable sampling procedure for the Woodroofe-type estimator for discrete
data over a finite number of possible values is therefore established.
Subsequently, the resulting vector of hazard rate estimators is proved to be
asymptotically normal with independent components. Asymptotic normality of the
survival function estimator is then established. Sister results for the
left-truncating random variable are also proved. Taken together, the resulting
joint vector of hazard rate estimates for the lifetime and left-truncation
random variables is proved to be the maximum likelihood estimate of the
parameters of the conditional joint lifetime and left-truncation distribution
given the lifetime has not been left-truncated. A hypothesis test for the shape
of the distribution function based on our asymptotic results is derived. Such a
test is useful to formally assess the plausibility of the stationarity
assumption in length-biased sampling. The finite sample performance of the
estimators is investigated in a simulation study. Applicability of the
theoretical results in an econometric setting is demonstrated with a subset of
data from the Mercedes-Benz 2017-A securitized bond.
