ON THE VALIDITY OF THE FORMAL EDGEWORTH EXPANSION FOR POSTERIOR DENSITIES

• 成果类型: Article
• 署名作者: Kolassa, John E.; Kuffner, Todd A.
• 署名单位: Rutgers University System; Rutgers University New Brunswick; Washington University (WUSTL)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1871
• 发表日期: 2020
• 页码: 1940-1958
• 关键词: asymptotic expansions; bartlett correction; approximations; distributions; cumulants

摘要

We consider a fundamental open problem in parametric Bayesian theory, namely the validity of the formal Edgeworth expansion of the posterior density. While the study of valid asymptotic expansions for posterior distributions constitutes a rich literature, the validity of the formal Edgeworth expansion has not been rigorously established. Several authors have claimed connections of various posterior expansions with the classical Edgeworth expansion, or have simply assumed its validity. Our main result settles this open problem. We also prove a lemma concerning the order of posterior cumulants which is of independent interest in Bayesian parametric theory. The most relevant literature is synthesized and compared to the newly-derived Edgeworth expansions. Numerical investigations illustrate that our expansion has the behavior expected of an Edgeworth expansion, and that it has better performance than the other existing expansion which was previously claimed to be of Edgeworth type.