We study a type of double truncation where units are observed if and only if their death event occurs within a specific timespan. The resulting missing data mechanism is nonignorable and thus has to be reconsidered. Based on the density function of observed lifetimes and the random sample size, we derive a likelihood model that enables simultaneous estimation of the lifetime distribution and the parameters governing the birth process. In particular, knowledge of the population size is not required. We show that the model is identifiable under certain conditions by using results on exponential families. Bayesian estimators and corresponding standard errors for all involved parameters become available by using MCMC simulation. We describe how the simulation can be performed efficiently while maintaining sufficiently good mixing behaviour of the resulting chains. Both finite-sample and asymptotic properties of the investigated estimators are examined through a simulation study. The proposed method is applied to estimate the lifetime distribution of German companies.

中文翻译：

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由时间限制数据收集引起的双重截断下寿命分布的贝叶斯估计

我们研究了一种双重截断，其中当且仅当它们的死亡事件发生在特定时间跨度内时才观察到单位。由此产生的缺失数据机制是不可忽略的，因此必须重新考虑。基于观察到的寿命的密度函数和随机样本大小，我们推导出一个似然模型，该模型能够同时估计寿命分布和控制出生过程的参数。特别是，不需要了解人口规模。我们通过使用指数族的结果表明该模型在某些条件下是可识别的。通过使用 MCMC 模拟，所有相关参数的贝叶斯估计量和相应的标准误差变得可用。我们描述了如何有效地执行模拟，同时保持所得链的足够良好的混合行为。通过模拟研究检查了所研究估计量的有限样本和渐近特性。所提出的方法用于估计德国公司的生命周期分布。