Abstract

We propose a framework for inference based on an “iterative likelihood function,” which provides a unified representation for a number of iterative approaches, including the EM algorithm and the generalized estimating equations (GEEs). The parameters are decoupled to facilitate construction of the inference vehicle, to simplify computation, or to ensure robustness to model misspecification and then recoupled to retain their original interpretations. For simplicity, throughout the paper, we will refer to the log-likelihood as the “likelihood.” We define the global, local, and stationary estimates of an iterative likelihood and, correspondingly, the global, local, and stationary attraction points of the expected iterative likelihood. Asymptotic properties of the global, local, and stationary estimates are derived under certain assumptions. An iterative likelihood is usually constructed such that the true value of the parameter is a point of attraction of the expected log-likelihood. Often, one can only verify that the true value of the parameter is a local or stationary attraction, but not a global attraction. We show that when the true value of the parameter is a global attraction, any global estimate is consistent and asymptotically normal; when the true value is a local or stationary attraction, there exists a local or stationary estimate that is consistent and asymptotically normal, with a probability tending to 1. The behavior of the estimates under a misspecified model is also discussed. Our methodology is illustrated with three examples: (i) estimation of the treatment group difference in the level of censored HIV RNA viral load from an AIDS clinical trial; (ii) analysis of the relationship between forced expiratory volume and height in girls from a longitudinal pulmonary function study; and (iii) investigation of the impact of smoking on lung cancer in the presence of DNA adducts. Two additional examples are in the supplementary materials, GEEs with missing covariates and an unweighted estimator for big data with subsampling. Supplementary files for this article are available online.

摘要

我们提出了一个基于“迭代似然函数”的推理框架，它为许多迭代方法提供了统一的表示，包括 EM 算法和广义估计方程 (GEE)。参数被解耦以促进推理工具的构建、简化计算或确保模型错误规范的鲁棒性，然后重新耦合以保留其原始解释。为简单起见，在整篇论文中，我们将把对数似然称为“似然”。我们定义了迭代似然的全局、局部和静止估计，相应地，定义了预期迭代似然的全局、局部和静止吸引点。全局、局部和平稳估计的渐近特性是在某些假设下导出的。通常构造迭代似然，使得参数的真实值是预期对数似然的吸引点。通常，人们只能验证参数的真实值是局部或静止的吸引力，而不是全局吸引力。我们表明，当参数的真实值是全局吸引力时，任何全局估计都是一致且渐近正态的；当真值是局部或平稳的吸引力时，存在一致且渐近正态的局部或平稳估计，概率趋于 1。还讨论了错误指定模型下估计的行为。我们的方法用三个例子来说明：(i) 估计治疗组在艾滋病临床试验中审查的 HIV RNA 病毒载量水平的差异；(ii) 纵向肺功能研究分析女孩用力呼气量与身高之间的关系；(iii) 在 DNA 加合物存在的情况下研究吸烟对肺癌的影响。补充材料中还有另外两个示例：缺少协变量的 GEE 和带有子采样的大数据的未加权估计量。本文的补充文件可在线获取。