Estimating a distribution function for discrete data subject to random truncation with an application to structured finance
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随机截断离散数据的分布函数估计及结构化金融应用
- 研究问题
- 金融产品期限有限、观测时间离散且数据受到随机左截断时，如何估计寿命分布及风险率，并对分布形状进行统计检验？
- 方法与思路
- 作者建立有限离散支持上的采样框架与 Woodroofe 型估计，推导风险率和生存函数的渐近分布，并构造分布形状检验。
- 材料陈述的贡献
- 摘要称建立寿命与截断变量相关风险率估计的渐近结果，证明联合风险率向量的条件最大似然性质，并以模拟和证券化债券数据演示。
- 阅读关注点
- 应核对左截断采样机制、寿命与截断变量的关系、离散支持限制，以及条件似然、分布形状检验与长度偏倚平稳性之间的联系。
- 阅读边界
- 这是 arXiv 预印本，同行评审或接收未核实；摘要未列完整采样假设和数值结果，证券化数据应用不能直接证明定价或风险预测能力。
原文摘要
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.