Empirical Bayes Confidence Intervals

The second installment of the Chamberlain (Virtual) Econometrics Seminar

featured Tim Armstrong presenting new work on Empirical Bayes Confidence

Intervals coauthored with  Michal Kolesár and Mikkel Plagborg-Møller,

and available from http://arxiv.org/abs/2004.03448.  Their approach seemed to

be motivated by recent work of Chetty using James-Stein shrinkage.  The

objective was to produce "robust" confidence intervals for such linear

shrinkage estimators, and the technique employed was a clever application

of the method of moment spaces.  This seemed to perform well in the

classical Gaussian sequence model when the mixing distribution was itself

Gaussian.  But as shown by Bruce Hansen in his discussion of the paper

following Tim's presentation, the procedure exhibits severe under

coverage in a typical Gaussian outlier model.  Thus, in the 1953 terminology

of Box it apparently possesses neither "robustness of validity", i.e. coverage,

nor "robustness of efficiency", i.e. length.

This led me to wonder how nonparametric empirical Bayes methods would

perform under such circumstances.  Bruce was kind enough to provide

code for his simulations and I spent the Easter weekend exploring this

question.  The result is yet another R Vinaigrette.  I compare two

flavors of nonparametric empirical Bayes Confidence intervals, both

based on what Efron calls G-modeling. A nonparametric estimate of the

mixing distribution called G serves as a "plug-in prior"  and intervals

are then based on posterior quantiles.  We consider two estimators for G:

the classical NPMLE of Kiefer and Wolfowitz, and the log-spline estimator

of Efron.  Spoiler alert:  in keeping with the samokritika (self-criticism)

spirit of the Vinaigrette -- Efron wins.  Links to the text and the code

for the simulations reported therein can be found at:

http://www.econ.uiuc.edu/~roger/research/ebayes/ebayes.html

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