Estimating a discrete distribution subject to random left-truncation with an application to structured finance
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 distrib...
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| Published in | Econometrics and statistics |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Elsevier B.V
01.06.2023
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| Online Access | Get full text |
| ISSN | 2452-3062 2452-3062 |
| DOI | 10.1016/j.ecosta.2023.05.005 |
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| 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. |
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| AbstractList | 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. |
| Author | Lautier, Jackson P. Yan, Jun Pozdnyakov, Vladimir |
| Author_xml | – sequence: 1 givenname: Jackson P. orcidid: 0000-0003-0294-0314 surname: Lautier fullname: Lautier, Jackson P. email: jlautier@bentley.edu organization: Department of Mathematical Sciences, Bentley University, Waltham, Massachusetts, USA – sequence: 2 givenname: Vladimir surname: Pozdnyakov fullname: Pozdnyakov, Vladimir organization: Department of Statistics, University of Connecticut, Storrs, Connecticut, USA – sequence: 3 givenname: Jun orcidid: 0000-0003-4401-7296 surname: Yan fullname: Yan, Jun organization: Department of Statistics, University of Connecticut, Storrs, Connecticut, USA |
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| Cites_doi | 10.3150/16-BEJ825 10.1080/01621459.1988.10478589 10.1016/0167-7152(95)00035-6 10.1198/016214502753479347 10.1007/BF00775812 10.1093/mnras/155.1.95 10.2307/2529588 10.1006/jmva.1993.1080 10.1093/biomet/asq069 10.1214/aos/1176346584 10.1214/aos/1024691085 10.1214/aos/1176349019 10.1017/S0269964800005064 10.1080/01621459.1991.10475011 10.1111/j.1467-9868.2010.00742.x 10.1214/aos/1176350826 10.1093/biomet/79.2.297 10.1111/j.1541-0420.2006.00710.x 10.1214/aos/1024691086 10.1214/aos/1176347991 10.1016/j.spl.2008.02.035 10.1093/biomet/asp026 10.1002/sim.2326 10.1214/aos/1176324529 10.1007/s10985-006-9012-2 10.1093/biomet/74.4.883 10.1007/BF02506482 10.1214/aos/1176345802 10.1093/biomet/64.2.225 10.1080/03610928708829561 10.1214/009053605000000372 10.1007/s00180-014-0496-z 10.1214/aos/1176350180 10.1006/jmva.1998.1806 10.1214/aos/1176347617 |
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| Keywords | consumer lease securitization reverse hazard rate Reg AB II asset-level disclosures asset-backed security product-limit estimator |
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| Title | Estimating a discrete distribution subject to random left-truncation with an application to structured finance |
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