Gaussian mixtures are commonly used for modeling heavy-tailed error distributions in robust linear regression. Combining the likelihood of a multivariate robust linear regression model with a standard improper prior distribution yields an analytically intractable posterior distribution that can be sampled using a data augmentation algorithm. When the response matrix has missing entries, there are unique challenges to the application and analysis of the convergence properties of the algorithm. Conditions for geometric ergodicity are provided when the incomplete data have a "monotone" structure. In the absence of a monotone structure, an intermediate imputation step is necessary for implementing the algorithm. In this case, we provide sufficient conditions for the algorithm to be Harris ergodic. Finally, we show that, when there is a monotone structure and intermediate imputation is unnecessary, intermediate imputation slows the convergence of the underlying Monte Carlo Markov chain, while post hoc imputation does not. An R package for the data augmentation algorithm is provided.
翻译:Gausian 混合物通常用于在强力线性回归中模拟重尾误差分布。 将多变量强线性回归模型的可能性与标准的先前不适当分布组合在一起, 会产生分析上棘手的后部分布, 可以使用数据增强算法进行抽样。 当响应矩阵缺少条目时, 对算法趋同特性的应用和分析会遇到独特的挑战。 当不完整数据有一个“ monoton” 结构时, 提供几何异性。 在没有单体结构的情况下, 执行算法需要一个中间估算步骤。 在这种情况下, 我们为算法提供了足够的条件, 使算法成为Harris ergodic。 最后, 我们显示, 当存在单体结构和中间估算时, 中间估算会减缓Monte Carlo Markov 链的趋同性, 而后期估算过程则不会。 提供了数据增强算法的 R 包 。