Propriety of the Posterior Distribution and Existence of the MLE for Regression Models With Covariates Missing at Random
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随机缺失协变量回归模型的后验正常化与极大似然估计存在性
- 研究问题
- 当回归协变量满足随机缺失且系数采用非正规均匀先验时，如何判断后验可正常化，并确认极大似然估计是否存在？
- 方法与思路
- 作者针对广义线性及右删失参数生存模型，推导可检查的协变量矩阵条件，将相关验证归结为线性方程组求解。
- 材料陈述的贡献
- 摘要称建立后验正常化和极大似然估计存在性的理论条件，涵盖缺失的分类与连续协变量，并给出大样本检查结果。
- 阅读关注点
- 应核对随机缺失和协变量分布假设、各条件是充分还是充要，以及使用全部样本与完整案例时后验性质为何可能不同。
- 阅读边界
- OpenAlex 摘要未注明稿件版本，未列出定理条件；后验正常化不等于参数识别，也不能据此保证任意缺失模型中的估计存在。
原文摘要
Characterizing model identifiability in the presence of missing covariate data is a very important issue in missing data problems. In this article, we characterize the propriety of the posterior distribution of the regression coefficients for some general classes of regression models, including the class of generalized linear models (GLM's) and parametric survival models with right-censored data. Toward this goal, we derive some very general and easy-to-check conditions for the matrix of covariates. We also derive sufficient conditions for the existence of the maximum likelihood estimates and establish novel results for checking propriety of the posterior when the sample size is large. Several theorems are given to establish propriety of the posterior and the existence of the maximum likelihood estimator. The conditions reduce to solving a system of linear equations, which can be carried out using software such as MAPLE, IMSL, or SAS. We assume that the missing covariates are missing at random and assume an improper uniform prior for the regression coefficients. In addition, we establish these results assuming a very general form for the covariate distribution, allowing for both missing categorical and/or continuous covariates. A small dataset is used to illustrate that the posterior can be improper based on complete cases while proper when all of the cases are used in the analysis. Two real datasets are presented to demonstrate verification of posterior propriety for GLM's and parametric survival models, and also to illustrate propriety for large datasets.