Interval-censored data may arise in questionnaire surveys when, instead of being asked to provide an exact value, respondents are free to answer with any interval without having pre-specified ranges. In this context, the assumption of noninformative censoring is violated, and thus, the standard methods for interval-censored data are not appropriate. This paper explores two schemes for data collection and deals with the problem of estimation of the underlying distribution function, assuming that it belongs to a parametric family. The consistency and asymptotic normality of a proposed maximum likelihood estimator are proven. A bootstrap procedure that can be used for constructing confidence intervals is considered, and its asymptotic validity is shown. A simulation study investigates the performance of the suggested methods.

中文翻译：

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具有信息间隔检查的调查数据的最大似然估计

问卷调查中可能会出现间隔审查的数据，这是因为被调查者可以自由地以任意间隔回答而无需事先提供确切的值，而不是要求提供确切的值。在这种情况下，违反了非信息检查的假设，因此间隔检查数据的标准方法不合适。本文探索了两种数据收集方案，并假设基础分布函数属于参数族，并处理了其估计问题。证明了所提出的最大似然估计的一致性和渐近正态性。考虑了可用于构造置信区间的自举程序，并显示了其渐近有效性。仿真研究调查了建议方法的性能。