TY - JOUR
T1 - Convergence analysis of data augmentation algorithms for Bayesian robust multivariate linear regression with incomplete data
AU - Li, Haoxiang
AU - Qin, Qian
AU - Jones, Galin L.
N1 - Publisher Copyright:
© 2024 Elsevier Inc.
PY - 2024/7
Y1 - 2024/7
N2 - 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.
AB - 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.
KW - Drift and minorization
KW - Geometric ergodicity
KW - Markov chain Monte Carlo
KW - Missing data imputation
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U2 - 10.1016/j.jmva.2024.105296
DO - 10.1016/j.jmva.2024.105296
M3 - Article
AN - SCOPUS:85184186990
SN - 0047-259X
VL - 202
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
M1 - 105296
ER -