Estimating a discrete distribution subject to random left-truncation with an application to structured finance

Abstract

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 is sampled from a financial trust, it will often be further subject to random left-truncation. The estimation of a distribution function from left-truncated data has been extensively addressed, but the case of discrete data over a known, finite number of possible values has not yet been thoroughly investigated. A precise discrete framework and suitable sampling procedure for the Woodroofe-type estimator for discrete data over a known, 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.