Robust Empirical Bayes Confidence Intervals
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We construct robust empirical Bayes confidence intervals (EBCIs) in a normal means problem. The intervals are centered at the usual empirical Bayes estimator, but use a larger critical value to account for the effect of shrinkage. We show that in this setting, parametric EBCIs based on the assumption that the means are normally distributed (Morris, 1983) can have coverage substantially below the nominal level when the normality assumption is violated, and we derive a simple rule of thumb for gauging the potential coverage distortion. In contrast, while our EBCIs remain close in length to the parametric EBCIs when the means are indeed normally distributed, they achieve correct coverage regardless of the means distribution. If the means are treated as fixed, our EBCIs have an average coverage guarantee: the coverage probability is at least 1-α on average across the n EBCIs for each of the means. We illustrate our methods with applications to effects of U.S. neighborhoods on intergenerational mobility, and structural changes in factor loadings in a large dynamic factor model for the Eurozone. Our approach generalizes to the construction of intervals with average coverage guarantees in other regularized estimation settings.
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