Although parametric empirical Bayes confidence intervals of multiple normal means are fundamental tools for compound decision problems, their performance can be sensitive to the misspecification of the parametric prior distribution (typically normal distribution), especially when some strong signals are included. We suggest a simple modification of the standard confidence intervals such that the proposed interval is robust against misspecification of the prior distribution. Our main idea is using well-known Tweedie's formula with robust likelihood based on $\gamma$-divergence. An advantage of the new interval is that the interval lengths are always smaller than or equal to those of the parametric empirical Bayes confidence interval so that the new interval is efficient and robust. We prove asymptotic validity that the coverage probability of the proposed confidence intervals attain a nominal level even when the true underlying distribution of signals is contaminated, and the coverage accuracy is less sensitive to the contamination ratio. The numerical performance of the proposed method is demonstrated through simulation experiments and a real data application.

翻译：虽然多种正常手段的参数实验性贝耶斯信任度间隔是复杂决策问题的基本工具,但其性能可以敏感地注意参数先前分布(通常正常分布)的偏差,特别是在包含一些强烈信号的情况下。我们建议简单修改标准信任度间隔,使拟议的间隔对先前分布的偏差具有稳健性。我们的主要想法是使用众所周知的Tweedie的公式,该公式的可靠可能性基于$\gamma$-digence。新间隔的一个优点是,间隔的长度总是小于或等于参数实验性海湾信任度间隔的长度,以便新的间隔有效和稳健。我们证明,即使真正的基本信号分布受到污染,拟议信任期的覆盖概率也达到名义水平,而且其准确度对污染率不那么敏感。通过模拟实验和真实的数据应用,可以证明拟议方法的数字性表现。