Rejoinder: Confidence Intervals for Nonparametric Empirical Bayes Analysis

Click to increase image sizeClick to decrease image size

Notes

1 We formally interpret ${\psi }^{*}$ as the Fourier transform of ψ, when $L(G)=\int \psi (\mu )dG(\mu )$ for some $\psi (·)$. This interpretation is valid as soon as $\psi ,\psi g$ and ${\psi }^{*}{g}^{*}\in {\mathbb{L}}_1(\mathbb{R})$. However, the result is applicable more generally. For example, the choice ${\psi }^{*}(t)=1$ corresponds to point evaluation of the density at 0, that is, $L(G)=g(0)$, see, Butucea and Comte (2009).

2 This “jump” as $\tau \to 0$ corresponds to a challenge in conducting inference for the local false sign rate that goes beyond the slow minimax rates, which hold for smooth classes such as (4).

3 As Pensky reminds us, the exact rate also depends on the choice of $\mathcal{G}$; in our applied examples the rate may be faster as we consider densities whose Fourier transform decays exponentially fast.

4 Here we follow Cressie’s terminology, who makes a distinction between “Bayes predictor” and “Bayes estimator.” Terminology varies across the literature, for example, Lee and Nelder (2009) write that “the word ‘prediction’ has often been used to denote the estimation of random effects. However, we believe that it is clearer to use prediction when we estimate future observations (unobservables) and estimation for the estimation of random effects in the data already observed.”

5 We provide code to reproduce the rejoinder’s analyses in the following Github repository: https://github.com/nignatiadis/empirical-bayes-confidence-intervals-paper.

6 The Anderson-Rubin construction underlying AMARI protects against loss of coverage when the denominator $f_G(z)$ of ${\theta }_G(z)$ is very small; however confidence intervals may get substantially wider when $f_G(z)\approx 0$. Monotonicity has played an important role in advancing empirical Bayes methodology (Houwelingen and Stijnen 1983; Koenker and Mizera 2014), and likewise, it would be important to modify AMARI in a way that enforces monotonicity. It seems challenging to do so; the ideas in Chernozhukov et al. (2010) may be useful for this goal.

7 This is often called the compound decisions problem, see, for example, Brown and Greenshtein (2009) and references therein.

8 Ignatiadis and Wager (2019) pursue this approach for James-Stein shrinkage with side-information X_i .