Nonparametric Empirical Bayes Simultaneous Estimation for Multiple Variances
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The shrinkage estimation has proven to be very useful when dealing with a large number of mean parameters in the normal models. In this paper, we consider the problem of simultaneous estimation of multiple variances and construct a shrinkage type estimator. We take the non-parametric empirical Bayes approach by starting with an arbitrary prior on the variances. Under an invariant loss function, the resultant Bayes estimator relies on the marginal cumulative distribution function of the sample variances. Replacing the marginal cdf by the empirical distribution function, we obtain a Non-parametric Empirical Bayes estimator for multiple Variances (NEBV). The proposed estimator converges to the corresponding Bayes version uniformly over a large set. Consequently, the NEBV works well in a post-selection setting. We then apply the NEBV to construct confidence intervals for mean parameters of the normal models in a post-selection setting. It is shown that the interval based on the NEBV is shortest among all the intervals which guarantee a desired coverage probability. Through real data analysis, we have further shown that the NEBV based intervals lead to the smallest number of discordances, a desirable property when we are faced with the current "replication crisis".
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