When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist) obtained from partial data valid? This paper answers this question by offering a new asymptotic theory about inference with missing data that is more general than existing theories. By using more powerful tools from real analysis and probability theory than those used in previous research, it proves that as the sample size increases and the extent of missingness decreases, the average-loglikelihood function generated by partial data and that ignores the missingness mechanism will almost surely converge uniformly to that which would have been generated by complete data; and if the data are Missing at Random, this convergence depends only on sample size. Thus, inferences from partial data, such as posterior modes, uncertainty estimates, confidence intervals, likelihood ratios, test statistics, and indeed, all quantities or features derived from the partial-data loglikelihood function, will be consistently estimated. They will approximate their complete-data analogues. This adds to previous research which has only proved the consistency and asymptotic normality of the posterior mode, and developed separate theories for Direct-Likelihood, Bayesian, and Frequentist inference. Practical implications of this result are discussed, and the theory is verified using a previous study of International Human Rights Law.
翻译:当从部分数据中获得推论(直接获益、巴耶斯或常识)时,何时从部分数据中获得推论(直接获益、巴耶斯或常识)是有效的?本文回答这一问题的方法是,提供一种新的关于与比现有理论更一般的缺失数据推论的无症状理论。通过使用比以往研究中更强大的实际分析和概率理论工具,它证明随着抽样规模的扩大和缺失程度的下降,由部分数据产生的平均比喻功能和忽略缺失机制几乎肯定会与完整数据产生的机制一致;如果数据在随机时缺失,这种趋同仅取决于抽样大小。因此,从部分数据(如远地点模式、不确定性估计、信任间隔、概率比率、测试统计,以及事实上,从部分数据对比值功能产生的所有数量或特征,将会得到一致的估计。它们将接近完整的数据类比。这与以前的研究相加起来,这些研究仅证明后端数据模式的一致性和正常性;如果数据在随机数据中缺少数据,这种趋同性则仅取决于样本的大小。因此,从部分理论和直位理论的后期研究中分别讨论了。