Asymptotic Properties for a Class of Partially Identified Models
一类部分识别模型的渐近性质AI 解读 ↓
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AI 根据所提供的论文材料归类：摘要核心是集合值随机变量下的一般部分识别推断；二元模型仅作为例子，证据不足以将整篇论文归入某个给定细分。
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一类部分识别模型的渐近性质
- 研究问题
- 当总体识别域可表示为集合值随机变量的 Aumann 期望的变换时，如何估计该识别域，并对整个集合及其子集开展统计推断？
- 方法与思路
- 作者将类比原则扩展至集合值随机变量，以 Minkowski 平均构造识别域的样本对应估计量，再用 Hausdorff 距离研究渐近分布与自助法。
- 材料陈述的贡献
- 摘要称识别域估计在 Hausdorff 距离下依概率收敛，并给出距离统计量的高斯过程极限、一致自助法，以及检验和两类置信集合族的构造。
- 阅读关注点
- 阅读全文时应核对集合值随机变量及识别域变换所需的正则条件，以及 Hausdorff 距离与有向距离检验如何对应两类置信集合族。
- 阅读边界
- 摘要中距离统计量的归一化因子缺失，无法确认具体收敛速度；适用条件与置信集合族的覆盖含义也需查阅全文，不能推广至任意部分识别模型。
原文摘要
We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include interval-identified scalar parameters, best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the population identification region with respect to the Hausdorff distance. We then show that the Hausdorff distance and the directed Hausdorff distance between the population identification region and the estimator, when properly normalized by , converge in distribution to functions of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate these limiting distributions. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and its subsets. We show that these results can be used to construct a confidence collection and a directed confidence collection. Those are (respectively) collection of sets that, when specified as a null hypothesis for the true value (a subset of values) of the population identification region, cannot be rejected by our tests.