A reverse Gaussian correlation inequality by adding cones

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• 作者：Xiaohong Chen^a ; ^{xiaohong.chen@yale.edu} ; Fuchang Gao^b ; ^{fuchang@uidaho.edu}
• 关键词：Gaussian measure ; Convex set ; Convex cone ; Posterior distribution ; Likelihood ratio ; Chi-bar distribution
• 刊名：Statistics & Probability Letters
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：123
• 期：Complete
• 页码：84-87
• 全文大小：369 K
• 卷排序：123

文摘

Let γγ denote any centered Gaussian measure on RdRd. It is proved that for any closed convex sets AA and BB in RdRd, and any closed convex cones CC and DD in RdRd, if D⊇C∘D⊇C∘, where C∘C∘ is the polar cone of CC, then γ((A+C)∩(B+D))≤γ(A+C)⋅γ(B+D),γ((A+C)∩(B+D))≤γ(A+C)⋅γ(B+D), and γ((A+C)∩(B−D))≥γ(A+C)⋅γ(B−D).γ((A+C)∩(B−D))≥γ(A+C)⋅γ(B−D). As an application, this new inequality is used to bound the asymptotic posterior distributions of likelihood ratio statistics for convex cones.