广义极大似然估计及其在分层抽样和多未观测层后分层中的应用
Generalized Maximum Likelihood Estimators and their applications to stratified sampling and post-stratification with many unobserved strata
摘要：
考虑估计$n$层均值的加权平均值的问题，该问题基于具有来自层$i, \; i=1,...,n$的已实现$K_i$观测的随机样本。在相当大一部分分层对应的$K_i=0$的情况下，该任务是重要的。这种情况可能发生在后分层中，当期望具有非常精细的分层时。可能需要精细分层，以便假设或近似值（如以分层为条件的随机缺失）具有吸引力。在观察性研究中，当希望通过对小而均匀的分层中的效应进行平均来估计平均治疗效应时，也可能需要精细分层。我们的方法是基于应用广义最大似然估计（GMLE），并与非参数经验贝叶斯的想法，以估计层$i$与相应的$K_i=0$的手段。没有关于未观察到的地层的平均值之间的关系的假设（即，与$K_i=0$）和观察地层的那些。我们的方法的性能证明在模拟和真实的数据集。一些一致性和渐近结果。此外，还给出了指数族混合均值的GMLE估计的相关基本结果。
英文摘要：
Consider the problem of estimating a weighted average of the means of $n$ strata, based on a random sample with realized $K_i$ observations from stratum $i, \; i=1,...,n$. This task is non-trivial in cases where for a significant portion of the strata the corresponding $K_i=0$. Such a situation may happen in post-stratification, when it is desired to have a very fine sftratification. A fine stratification could be desired in order that assumptions, or, approximations, like Missing At Random conditional on strata, will be appealing. A fine stratification could also be desired in observational studies, when it is desired to estimate average treatment effect, by averaging the effects in small and homogenous strata. Our approach is based on applying Generalized Maximum Likelihood Estimators (GMLE), and ideas that are related to Non-Parametric Empirical Bayes, in order to estimate the means of strata $i$ with corresponding $K_i=0$. There are no assumptions about a relation between the means of the unobserved strata (i.e., with $K_i=0$) and those of the observed strata. The performance of our approach is demonstrated both in simulations and on a real data set. Some consistency and asymptotic results are also presented. In addition, related basic results about GMLE estimation of the mean of mixtures of exponential families are provided.
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