text:
\section{Introduction, model, and proposed method}

Consider a scalar Gaussian location model \[
Y_i \mid \theta_i, \sigma_i^2 \sim \Norm(\theta_i, \sigma_i^2) \quad i = 1,\ldots, n
\]
where $(Y_i, \theta_i, \sigma_i^2) \iid P$ are drawn from some superpopulation $P$. Assume
that the data analyst observes $(Y_i, \sigma_i^2)$ and is interested in estimating the
unknown and random parameters $\theta_1,\ldots, \theta_n$. Consider the compound Bayes
risk for separable decision rules $\delta_i(y) \equiv \delta(y, \sigma_i)$, for a fixed
set of $\sigma_i^2$: \[
\B_n(\{\delta_i\}; \sigma_{1:n}^2) = \E\bk{
    \frac{1}{n}\sum_{i=1}^n (\delta(Y_i; \sigma_i) - \theta_i)^2 \mid \sigma_1^2,\ldots,
    \sigma_n^2
}.
\]
The optimal decision rules take the form of posterior means $\delta^\star(Y_i;
\sigma_i) = \E[\theta_i \mid Y_i,
\sigma_i^2]$. Indeed, we can view $\theta_i \mid \sigma_i \sim G_{(i)}$ as a
prior for $\theta_i$ and
$Y_i \mid \theta_i, \sigma_i \sim \Norm(\theta_i, \sigma_i^2)$ as a Gaussian location
signal; as a result, $\delta^\star(Y_i; \sigma_i)$ is the posterior mean. However,
$\delta^\star$ requires knowledge of $G_{(i)}$, which is unknown in practice. We shall refer
to $\delta^\star$ as the oracle Bayesian posterior means.

Empirical Bayes
\citep{robbins1956,morris1983parametric,efron1973stein,efron2011tweedie,efron2019bayes} is
the strategy of mimicking the decision rules $\delta^\star$ by estimating $G_{(i)}$ from the
data $(Y_i, \sigma_i)_{i=1}^n$. For a function $h(Y, \theta)$, a prior distribution $G$,
and $\sigma > 0$, let\footnote{We let
$\varphi(t)=\frac{1}{\sqrt{2\pi}} e^{-t^2/2}$ be the PDF of the Gaussian distribution.}
\[\mathbf{E}_{G,
\sigma}[h(Y, \theta) \mid y] = \frac{\int h(y, \theta) \varphi\pr{\frac{y-\theta}{\sigma}}
\frac{1}
{\sigma} G(d\theta)}{\int \varphi\pr{\frac{y-\theta}{\sigma}}\frac{1} {\sigma}
G(d\theta)}\] be the posterior mean of $h(Y, \theta)$ evaluated at $Y=y$, for the normal
location model $Y \mid \theta, \sigma^2 \sim \Norm(\theta, \sigma^2)$ and $\theta \mid
\sigma^2 \sim G$. Given an estimate $\hat G_{(i)}$ of the prior $G_{(i)}$, empirical Bayes
methods---specifically, those in the \emph{$g$-modeling} tradition\footnote{Alternatively,
by \emph{Tweedie's formula}, \[
\mathbf{E}_{G_{(i)}, \sigma_i}[\theta \mid Y=y] = y + \sigma_i^2\frac{f'_{G_{(i)}, \sigma_i}(y)}
{f_{G_{(i)}, \sigma_i}(y)}
\]
for marginal distribution $f_{G,\sigma}(y) = \int \frac{1}{\sigma}\varphi\pr{
\frac{y-\theta}{\sigma}}\,G(d\theta)$. Empirical Bayes methods in the \emph{$f$-modeling}
tradition attempts to estimate $f_{G_{(i)}, \sigma_i}$ and plug in the estimated $f$ to
Tweedie's formula to form an estimate of the oracle Bayesian's posterior mean. }
\citep{efron2019bayes}---often plug in $\hat G_{(i)}$ and use the estimated posterior mean
\[
\delta_{\mathrm{EB}}(y, \sigma_i; \hat G_{(i)}) = \mathbf{E}_{\hat G_{(i)}, \sigma_i}\bk{\theta \mid y},
\]
in the hopes that the \emph{expected regret} against the oracle Bayesian---that is, the
difference
between the empirical Bayesian's Bayes risk and that of the oracle Bayesian--- \[
\B_n(\{\delta_{\mathrm{EB}}(\cdot, \sigma_i; \hat G_{(i)})\}; \sigma_{1:n}^2) - \B_n(\{\delta^\star(\cdot,
\sigma_i)\}; \sigma_{1:n}^2) \numberthis \label{eq:expected_regret_formal}
\]
is small.

However, estimating $G_{(i)}$ can be difficult. After all, for each $G_{(i)}$, we observe
a single draw $Y_i \sim G_{(i)} \star \Norm(0,\sigma_i^2)$ from the convolution of
$G_{(i)}$ and a Gaussian distribution. Without somehow pooling across different $Y_i$'s,
it is impossible to estimate $G_{(i)}$ reliably. To avoid these difficulties, existing
literature \citep{jiang2020general,soloff2021multivariate} often assume that $G_{(i)} =
G_{(0)}$ is the same across $i$---in other words, $\theta_i \indep \sigma_i^2$ under
$P$---and proposes estimating $G_{(0)}$ via nonparametric maximum likelihood
\citep[NPMLE,][]{kiefer1956consistency,koenker2014convex,jiang2009general}.

While convenient, such an independence assumption is often rejected in many economic
contexts \citep[e.g., the Opportunity Atlas,][]{chetty2018opportunity}. In many empirical
settings in economics where empirical Bayes methods are appropriate, $Y_i$ is an estimate
of some parameter $\theta_i$, and $\sigma_i$ is the associated standard error for $Y_i$.
The normality of $Y_i$ is motivated by the asymptotic approximation from an underlying
micro-data sample of size $n_i$ that generates $Y_i$:\footnote{Of course, the standard
error is usually estimated and not known. That is, we typically have the Gaussian
approximation
\[
\sqrt{n_i} (Y_i - \theta_i) \dto \Norm(0, \sigma_{0i}^2)
\]
and an estimated standard error of the form $\hat \sigma_i^2 = \hat \sigma_{0i}^2 / n_i$.
But since $\hat \sigma_{0i}^2$ is usually $\sqrt{n_i}$-consistent for $\sigma_{0i}^2$, the
uncertainty in $\hat\sigma_i^2$ is much smaller than that in $Y$, due to the scaling by
$n_i$. As a result, we treat $\sigma_i^2$ as known. } \[\frac{(Y_i - \theta_i)}{\sigma_i}
\dto \Norm(0, 1) \text{ as the underlying sample size $n_i \to \infty$.}\] For instance,
in the Opportunity Atlas, $\theta_i$ may be \emph{economic mobility}, defined, say,  as
the mean income rank as adults of those growing up in Census tract $i$, whose parents'
income is at the 25\th{} percentile nationally. $(Y_i, \sigma_i)$ are then estimates and
standard errors for economic mobility in Census tract $i$. For tracts where $\sigma_i$ is
larger---either because the variability of adult income is high or because the there are
not many individuals whose parents' incomes are at the 25\th{} percentile---we may expect
these tracts to be more or less economically mobile. That is, we should expect $\sigma_i$
to predict $\theta_i$: i.e., they are not independent. Indeed, the dependence is apparent
in \cref{fig:kfr_pooled_raw_data} for the raw estimates, and to a smaller extent for the
estimates residualized on a vector of tract-level characteristics. Such dependence is more
pronounced for other measures of economic mobility---as we show, for instance, in
\cref{fig:kfr_top20_black_raw_data,fig:jail_black_raw_data}, for two binary outcomes.
Posterior means from empirical Bayes methods that erroneously assume independence may then
underperform.

\begin{figure}[htb]
  \centering
  \includegraphics[width=\textwidth]{assets/kfr_pooled_raw_data.png}
  \caption{Scatter plot of $Y_i$ against $\log_{10}(\sigma_i)$ for Opportunity Atlas
  measure of mean family income rank of an individual growing up in tract $i$ whose
  parents are at the 25\th{} national family income percentile. All tracts within the
  largest 20 Commuting Zones (CZs) are shown. The right panel plots the scatter plot for
  $Y_i$ defined as the estimates residualized linearly by a vector of covariates. The
  covariates used are a constant term, poverty rate in 2000, share of Black individuals
  in 2010, mean
  household income in 2000, log wage growth for high school graduates, mean family income
  rank of parents, mean family income rank of black parents, and the fraction with college
  or post-graduate degrees in 2010. The residualization weights by the precision
  $1/\sigma_i^2$. The red dashed line plots estimates of $\E[Y \mid \sigma] \pm 2
  \sqrt{\var(Y\mid
  \sigma)}$ under the assumption that $\theta_i \indep \sigma_i$.}
  \label{fig:kfr_pooled_raw_data}
\end{figure}

\begin{figure}[htb]
  \centering
  \includegraphics[width=\textwidth]{assets/kfr_top20_black_raw_data.png}
  \caption{Scatter plot of $Y_i$ against $\log_{10}(\sigma_i)$ for Opportunity Atlas
  measure of the probability of family income ranking in the national top 20\%, for a
  Black individual growing up in tract $i$ whose parents are at the 25\th{} national
  family income percentile. All tracts within the largest 20 Commuting Zones (CZs) are
  shown. The residualization is with the same specification as in
  \cref{fig:kfr_pooled_raw_data}. Due to the regression specification in
  \cite{chetty2018opportunity}, point estimates of this quantity do not always lie within
  $ [0,1]$. The red dashed line plots estimates of $\E[Y \mid \sigma] \pm 2
  \sqrt{\var(Y\mid
  \sigma)}$ under the assumption that $\theta_i \indep \sigma_i$.}
  \label{fig:kfr_top20_black_raw_data}
\end{figure}

\begin{figure}[htb]
  \centering
  \includegraphics[width=\textwidth]{assets/jail_black_raw_data.png}
  \caption{Scatter plot of $Y_i$ against $\log_{10}(\sigma_i)$ for Opportunity Atlas
  measure of the Black male incarceration rate among those whose family is at the 25\th{}
  percentile. All tracts within the largest 20 Commuting Zones (CZs) are
  shown. The residualization is with the same specification as in
  \cref{fig:kfr_pooled_raw_data}. Due to the regression specification in
  \cite{chetty2018opportunity}, point estimates of this quantity do not always lie within
  $ [0,1]$. The red dashed line plots estimates of $\E[Y \mid \sigma] \pm 2
  \sqrt{\var(Y\mid
  \sigma)}$ under the assumption that $\theta_i \indep \sigma_i$.}
  \label{fig:jail_black_raw_data}
\end{figure}

Motivated by these issues, we propose the following assumption: The conditional
distribution of $\theta_i \mid \sigma_i$ follows a location-scale family with shape $G_0$,
location parameter $m_0(\sigma_i)$, and scale parameter $s_0(\sigma_i)$: \[
\P(\theta_i \le t \mid \sigma_i) = G_0\pr{\frac{t - m_0(\sigma_i)}{s_0(\sigma_i)}}
\numberthis \label{eq:location_scale}
\]
If $G_0$ has at least two moments, without loss of generality, the location and scale
parameters are conditional moments and identifiable from the joint distribution $(Y_i,
\sigma_i^2)$: \begin{align*}
m_0(\sigma_i) & = \E[\theta_i \mid \sigma_i] = \E[Y_i \mid \sigma_i] \\
s_0^2(\sigma_i) & = \var(\theta_i \mid \sigma_i ) = \var(Y_i \mid \sigma_i) - \sigma_i^2.
\end{align*}
The above parametrization implies the normalization that $G_0$ has zero mean and unit
variance. Under \eqref{eq:location_scale}, define the transformed data and parameters by
subtracting the conditional mean $m_0$ and normalizing by the conditional standard
deviation $s_0$:\[ Z_i = \frac{Y_i - m_0(\sigma_i)}{s_0(\sigma_i)} \quad \tau_i =
\frac{\theta_i - m_0 (\sigma_i)}{s_0(\sigma_i)} \quad \nu_i =
\frac{\sigma_i}{s_0(\sigma_i)}.
\]
Then, thanks to \eqref{eq:location_scale}, the transformed data $(Z_i, \tau_i, \nu_i)$ do follow a
heteroskedastic Gaussian
location model for which independence $\tau_i \indep \nu_i$: \[
Z_i \mid \tau, \nu_i^2 \sim \Norm(0, \nu_i^2) \quad \tau_i \mid \nu_i^2 \sim G_0.
\numberthis\label{eq:transformed_gaussian}
\]

Motivated by \eqref{eq:transformed_gaussian}, we propose two empirical Bayes methods. Both
require nonparametric estimators $\hat m$ of $m_0$ and $\hat s$ of $s_0$ from the data $
(Y_i, \sigma_i)$. We denote the nuisance parameters and their estimates as $\hat\eta =
(\hat m, \hat s)$ and $\eta_0 = (m_0, s_0)$. Correspondingly, define the estimated
transformed data \[
\hat Z_i = \frac{Y_i - \hat m(\sigma_i)}{\hat s(\sigma_i)} \quad \hat\nu_i = \frac{\sigma_i}
{\hat s(\sigma_i)}. \numberthis \label{eq:transformed_data}\]
Both methods perform empirical Bayes analysis $\tau_i$ on the data $(\hat Z_i,
\hat\nu_i)$. They then transform posterior means for $\tau_i$ into posterior means for
$\theta_i$ by recognizing that \[
\E\bk{\theta_i \mid Y_i, \sigma_i} = m_0(\sigma_i) + s_0(\sigma_i) \E[\tau_i \mid Z_i,
\nu_i]. \numberthis \label{eq:relation_posterior_mean}
\]

Our first method is motivated by assuming that $G_0$ is standard Gaussian. If $G_0 \sim
\Norm(0,1)$, then the corresponding posterior mean for $\tau_i$ is known in closed form:
\[
\mathbf{E}_{\Norm(0,1), \hat\nu_i}[\tau_i \mid \hat Z_i] = \frac{1}{1 + \hat\nu_i^2} \hat Z_i =
\frac{\hat s^2(\sigma_i)}{\hat s^2(\sigma_i) + \sigma_i^2} \hat Z_i
\]
and thus the posterior mean estimate is \[
\hat\theta_{i, \Norm(0,1), \hat\eta} = \hat m(\sigma_i) + \frac{\hat s^2(\sigma_i)}{\hat s^2
(\sigma_i) + \sigma_i^2} (Y_i - \hat m(\sigma_i)). \numberthis \label{eq:gaussian_cond_eb}
\]
The population analogue for \eqref{eq:gaussian_cond_eb} is \[
\hat\theta_{i, \Norm(0,1), \eta_0} = m_0(\sigma_i) + \frac{s_0^2(\sigma_i)}{s_0^2
(\sigma_i) + \sigma_i^2} (Y_i - m_0(\sigma_i)). \numberthis \label{eq:gaussian_cond_b}
\]
In turn, \eqref{eq:gaussian_cond_b} is the posterior mean under the distributional
assumption that $\theta_i \mid \sigma_i$ is Gaussian. However, \eqref{eq:gaussian_cond_b}
also admits a nonparametric interpretation as the optimal (i.e., Bayes
risk $\B_n$-minimizing) \emph{linear}
decision rule of the
form \[
\delta(Y, \sigma) = a(\sigma)  + b(\sigma) Y,
\]
which is linear in $Y$ \citep{weinstein2018group}.

Our second method estimates $G_0$ nonparametrically via NPMLE. Specifically, let $\mathcal
P(\R)$ denote the set of all real-valued distributions. We consider an NPMLE for
\eqref{eq:transformed_gaussian}, replacing the unknown $Z_i, \nu_i$ with their empirical
counterparts:
\[
\hat G_n \in \argmax_{G \in \mathcal P(\R)} \frac{1}{n} \sum_{i=1}^n \log \pr{\int_
{-\infty}^\infty \varphi
\pr{
\frac{\hat Z_i - \tau}{\hat\nu_i}} \,G(d\tau)}. \numberthis \label{eq:npmle}
\]
Corollary 3 in \cite{soloff2021multivariate} shows that all solutions to \eqref{eq:npmle}
are discrete distributions supported within the range of the data $[\min_i \hat Z_i,
\max_i \hat Z_i]$. In practice, we approximate\footnote{Our theoretical results allow for
approximate NPMLEs, and thus deals with the approximation. We also note that the
population analogue of \eqref{eq:npmle} is uniquely maximized by $G_0$, but it is unknown
whether the sample problem \eqref{eq:npmle} has a unique solution with probability one, in
settings with heteroskedasticity.} $\mathcal P(\R)$ with all distributions over a fine
grid of values within $[\min_i \hat Z_i,
\max_i \hat Z_i]$, so that the resulting optimization problem is finite dimensional,
concave, and efficiently computed via modern convex optimization methods implemented in
\textsf{REBayes} \citep{koenker2014convex,koenker2017rebayes}.\footnote{None of the
procedures we propose are computationally intensive. For our numerical experiments in
\cref{sec:empirical}, with 10,000 data points, our procedures (along with sampling
simulated data and residualizing against covariates) take less than 30 seconds on a 2022
Apple Mac Studio (Apple M1 Max, 32GB RAM) without explicit parallelization.} Given $\hat
G_n,$ we can compute the estimated posterior means \footnote{Here, we slightly abuse
notation and write $\mathbf{E}_{G,
\nu}[\tau \mid Z]$ as the posterior mean for $\tau$ evaluated at $Z$, rather than that of
$\theta$ at $Y$.} \[
\hat\theta_{i, \hat G_n, \hat\eta} = \hat m(\sigma_i) + \hat s(\sigma_i) \mathbf{E}_{\hat G_n,
\hat\nu_i}[\tau_i \mid \hat Z_i].
\]

To summarize, the methods we propose follow these two steps:
\begin{enumerate}
  \item Nonparametrically estimate $m_0(\sigma) = \E[Y \mid \sigma], s_0^2(\sigma) = \var
  (Y \mid \sigma) - \sigma^2$. Both are feasible with nonparametric regression methods
  (e.g. local polynomial regression).
  \item With the estimators $\hat m, \hat s$, transform the data according to
  \eqref{eq:transformed_data}. Then either
  \begin{enumerate}
    \item Report the optimal linear decision rules \eqref{eq:gaussian_cond_eb}, or
    \item Estimate $\hat G_n$ through \eqref{eq:npmle} (say, via \texttt{REBayes::GLMix}
    in \textsf{R}). Transform the posterior means for $\tau$ back into those for $\theta$
    via
    \eqref{eq:relation_posterior_mean}.
  \end{enumerate}
\end{enumerate}

The rest of this note is structured as follows. \Cref{sec:regret} introduces our main
result on the regret properties of $\hat\theta_{i, \hat G_n, \hat\eta}$, as well as results
linking regrets of different economic objects to regret in mean-squared error.
\Cref{sec:empirical} concludes the paper with an empirical illustration calibrated to the
Opportunity Atlas. All proofs are deferred to the Appendix.

In what follows, we may use the symbol $C$ to denote a generic positive and finite
universal constant. We may use the symbol $C_{x}$ to denote a generic positive and finite
universal constant that depends only on $x$, some parameter(s) that describe the problem.
Occurrences of the same symbol $C, C_x$ may not refer to the same constants. Similarly,
for $A,B \ge 0$, we use $A \lesssim B$ to mean that some universal $C$ exists such that $A
\le C B$, and we use $A
\lesssim_x B$ to mean that some universal $C_x$ exists such that $A \le C_x B$. We say $A
\rateeq B$ if $A \lesssim B$ and $B \lesssim A$. In logical statements, appearances of
$\lesssim$ implicitly prepend ``there exists a universal constant'' to the statement. For
instance, statements like ``under certain assumptions, $\P(A \lesssim B) \ge c_0$'' should
be read as ``under certain assumptions, there exists a universal constant $C>0$ such that
$\P(A \le CB) \ge c_0$.''

\subsection{Related literature} The literature in empirical Bayes methods is vast. For
theoretical and applied results of particular interest to economists, see the recent
lecture by \cite{guwalters_eb} and references therein. The \emph{Statistical Science}
article by \cite{efron2019bayes} and its discussions and rejoinders are excellent
introductions to the statistics literature on empirical Bayes. Here, we highlight a few
papers in the econometrics and statistics literature that discuss similar issues.

We are not the first to identify the challenge heteroskedasticity (and non-independence
between $\theta_i$ and $\sigma_i$) poses to empirical Bayes analysis, nor are we the first
to offer potential solutions. Earlier work by \cite{weinstein2018group} propose
\eqref{eq:gaussian_cond_eb} by estimating $m_0, s_0$ with nearest-neighbor methods. When
microdata (the data that generate the estimate $Y_i$) are available,
\cite{gu2017empirical} show that if the microdata themselves have a Gaussian distribution,
then one can treat the mean estimate $Y_i$ and the variance estimate $\sigma_i^2$ as
following a bivariate exponential family distribution and apply the NPMLE. In the same
model, \cite{banerjee2020nonparametric} propose an $f$-modeling approach.\footnote{A
version of the non-independence problem also manifests when microdata is available. See
\cref{rmksq:nonindep}.}
\cite{kwon2021optimal} and \cite{xie2012sure} also consider a heteroskedastic normal
location model. Their risk criterion is a frequentist one, where $\theta_{1:n}$ are
assumed to be fixed (or conditioned upon) rather than sampled from some superpopulation.
They use Stein's
Unbiased Risk Estimate to select hyperparameters for over a particular class of shrinkage
procedures.

Our theoretical results hew closely to---and extend---the regret results in
\cite{jiang2020general} and \cite{soloff2021multivariate}, which themselves are extensions
of earlier results in the homoskedastic setting
\citep{jiang2009general,saha2020nonparametric}. These results show that empirical Bayes
derived from estimating the prior $G$ via NPMLE achieve near-oracle performance, in the
sense that the regret against the oracle Bayesian converges to zero at appreciable rates.
We extend these results by characterizing the effect of an estimated nuisance parameter in
a first step, before estimating the NPMLE.

Finally, our location-scale assumption \eqref{eq:location_scale} bears resemblance to how
covariates are used in \cite{ignatiadis2019covariate}. They assume a homoskedastic setting
where the prior depends on some covariates $X_i$: i.e., in our notation, $\theta_i
\mid X_i
\sim \Norm(m(X_i), \sigma^2_0)$ and $Y_i \mid \theta_i \sim \Norm (\theta_i, \sigma^2)$.
We obtain \eqref{eq:location_scale} if (i) we make the setting heteroskedastic, (ii) set
$X_i =
\sigma_i^2$, (iii) generalize the prior shape to $G_0$, and (iv) assume the prior variance
also depends on $X_i = \sigma_i^2$.

\section{Regret control}
\label{sec:regret}

Having restricted ourselves to empirical Bayes methods via $g$-modeling under the
location-scale model \eqref{eq:location_scale}, let us define \[
\mathrm{MSERegret}_n(G, \eta) = \frac{1}{n} \sum_{i=1}^n (\hat\theta_{i, G, \eta} - \theta_i)^2 -
\frac{1}{n} \sum_{i=1}^n \pr{\theta_i^* - \theta_i}^2 \quad \theta_i^* \equiv \hat\theta_
{i, G_0, \eta_0}
\]
as the excess loss of using prior $G$ and nuisance parameter estimate $\eta$, relative to
the oracle Bayesian's posterior mean.

From this point onwards, we treat
$\sigma_1,\ldots,\sigma_n$ as fixed (by conditioning on them), and all expectation or
probability statements are with respect to $\theta_i \mid \sigma_i$ under
\eqref{eq:location_scale}.Now, we note that the expected value $\E[\mathrm{MSERegret}_n(\hat G_n, \hat\eta)]$ equals the expected
regret
\eqref{eq:expected_regret_formal} for posterior means derived from the NPMLE $\hat G_n$.

We assume $n > 15$. We also assume that $\hat G_n$ is an approximate maximizer of the
log-likelihood on the transformed data $\hat Z_i$ and $\hat\nu_i$. Let $\psi_i(Z_i,
\hat\eta, G) = \log \pr{\int
\varphi
\pr{\frac{\hat Z_i -\tau}{\hat \nu_i}} G(d\tau)}$ be the objective function in
\eqref{eq:npmle}. Then, we assume that $\hat G_n$ satisfies \[
\frac{1}{n} \sum_{i=1}^n \psi_i(Z_i, \hat\eta, \hat G_n) \ge \sup_{H \in \mathcal P(\R)}
\frac{1}{n} \sum_{i=1}^n \psi_i(Z_i, \hat\eta, H) - \kappa_n
\numberthis \label{eq:approx_mle}
\]
for tolerance $\kappa_n$ set to \[\kappa_n = \frac{2 \log \pr{\frac{n}{
\sqrt{2\pi} e}}}{n} \numberthis
\label{eq:kappa_def}.\] Moreover, we
require that $\hat G_n$ has support points within $[\min_i\hat Z_i,
\max_i \hat
Z_i]$. This assumption is not restrictive, since all maximizers of the
likelihood function satisfy this constraint \citep[See Corollary 3 in][]{soloff2021multivariate}.

We now state further assumptions on the data-generating process $P$ beyond \eqref{eq:location_scale}.

\begin{restatable}{as}{moments}
\label{as:moments}
The distribution $G_0$ is has mean zero, unit variance, and admits simultaneous moment
control with parameter $\alpha \in (0,2]$: There exists a constant $A$ such that \[
\mu_p(G_0) \equiv \br{\E_{\tau \sim G_0}[|\tau|^p]}^{1/p} \le A_0 p^
{1/\alpha}. \numberthis \label{eq:pthmomentdef}
\]
\end{restatable}

\Cref{as:moments} states that the distribution $G_0$ has thin tails, where the parameter
$\alpha$ controls how thin its tails are. An equivalent statement is that there exists
$a_1, a_2$ such that $\P_{G_0}(|\tau| > t) \le a_1\exp\pr{-a_2 t^\alpha}$ for all $t > 0$.
Note that
when $\alpha = 2$, $G_0$ is subgaussian, and when $\alpha = 1$, $G_0$ is subexponential.

\begin{restatable}{as}{variancebounds}
\label{as:variance_bounds}
The variances \[
\sigma_\ell \le \sigma_i \le \sigma_u \quad s_\ell \le s_{0i} \le s_u
\]
admit lower and upper bounds $0 < \sigl, \sigu, s_{0\ell}, s_{0u} < \infty$. This implies that $0 <
\nu_\ell \le \nu_i \le \nu_u < \infty$.
\end{restatable}

\Cref{as:variance_bounds} bounds the variance parameters away from zero and infinity. This
is a standard assumption in the literature, maintained likewise by
\cite{jiang2020general} and \cite{soloff2021multivariate}.

\begin{restatable}{as}{holder}
\label{as:holder}
Let $C_{A_1}^p([\sigma_\ell,\sigma_u])$ be the H\"older class of order $p \in \N$ with maximal
H\"older norm $A_1 > 0$ supported on $[\sigma_\ell,\sigma_u]$.\footnote{See Section 2.7.1 of \cite{vaart1996weak} for the
definition.} Then $m_0, s_0 \in C_{A_1}^p([\sigma_\ell,\sigma_u])$.
We let $\mathcal H$ collect the set of hyperparameters of the problem setup $\mathcal{H}
= (\sigma_\ell,
\sigma_u, s_\ell, s_u, A_0, A_1, \alpha, p)$.

The estimators for $m_{0}$ and $s_0$, $\hat\eta = (\hat m, \hat s)$, satisfy that, for some
$\beta_0 > 0$,
\begin{enumerate}
    \item $\E[\norm{\hat m - m_{0}}_\infty] \lesssim_{\mathcal{H}} n^{-\frac{p}
    {2p+1}} (\log n)^{\beta_0}$
    \item  $\E[\norm{\hat s - s_{0}}_\infty] \lesssim_{\mathcal{H}} n^{-\frac{p}
    {2p+1}} (\log n)^{\beta_0}$
    \item $\P\bk{\hat m, \hat s \in C_{A_1}^p([\sigma_\ell,\sigma_u])} = 1$
    \item $\P\bk{s_{0\ell} < \hat s < s_{0u}} = 1$.
\end{enumerate}
\end{restatable}

\Cref{as:holder} is a smoothness assumption on the nuisance parameters $m_0$ and $s_0$.
Moreover, it is also a high-level assumption on the quality of the estimation procedure
for $\hat m, \hat s$. Specifically, \cref{as:holder} expects that the nuisance parameter
estimates $\hat m$ and $\hat s$ are rate-optimal up to logarithmic factors---since the rate
$n^{-\frac{p}{p+1}} (\log n)^{\beta_0}$ matches the form of the sup-norm rate in
\cite{stone1980optimal}. The assumption also expects the nuisance parameter estimates to
live in the same smoothness class as the unknown parameters\footnote{By the triangle
inequality, projecting a preliminary estimate $\tilde \eta$ to the smoothness class in
$\norm{\cdot}_\infty$ incurs, at most, twice the estimation error. That
is, let $\hat\eta$ be an element in the H\"older class that minimizes $\norm{\hat\eta -
\tilde\eta}_\infty.$ Then, $
\norm{\hat\eta - \eta_0}_\infty \le \norm{\hat\eta - \tilde\eta}_\infty + \norm{\eta_0 -
\tilde\eta}_\infty \le 2 \norm{\eta_0 - \tilde\eta}_\infty. $} and to respect the
boundedness constraints for $s_0$.

Consider the event, indexed by a constant $C_1$, \[\mathbf{A}_n(C_1) \equiv
\br{\max\pr{\norm{\hat s - s_0}_\infty , \norm{\hat m - m_0}_\infty} \le C_1
n^{-\frac{p}{2p+1}} (\log n)^{\beta_0}}.
\numberthis \label{eq:bold_A_def}
\] $\mathbf{A}_n$ indicates that the nuisance parameter
estimates satisfy some rate in $\norm{\cdot}_\infty$. Note that, by \cref{as:holder} and
Markov's inequality, there exists choices of $C_1$ so that $\P[\mathbf{A}_n(C_1)]$ is arbitrarily
close to 1. Our main result derives a convergence rate for the expected MSE regret
conditional on the event $\mathbf{A}_n(C_1).$

\begin{restatable}{theorem}{maintext}
\label{thm:maintext}
Suppose $n > 15$. Assume \cref{as:holder,as:moments,as:variance_bounds} hold. Suppose
$\hat G_n$ satisfy \eqref{eq:approx_mle} and have support within $[\min_i \hat Z_i, \max_i
\hat Z_i]$. Then, for any $\delta \in (0, \frac{1}{2})$, there exists universal constants $C_{1,
\mathcal{H}, \delta} > 0$ and $C_{0, \mathcal{H}} > 0$ such that $\P(\mathbf{A}_n(C_{1,
\mathcal{H},
\delta})) \ge 1-\delta$ and that the expected regret conditional on $\mathbf{A}_n$ is dominated by the rate function \[
\E\bk{
    \mathrm{MSERegret}_n(\hat G_n, \hat\eta)
 \mid \mathbf{A}_n(C_{1, \mathcal{H}, \delta})} \le C_{0, \mathcal{H}} n^{-\frac{2p}{2p+1}} (\log n)^{\frac{2+\alpha}{\alpha} + 3 + 2\beta_0}.
 \numberthis \label{eq:regret_rate_final}
\]
\end{restatable}

If the event $\mathbf{A}_n(C_1)$ is sufficiently likely, we can control expected regret on the
complement $\mathbf{A}^\mathrm{C}_n$ as well.

\begin{restatable}{cor}{cormaintext}
\label{cor:maintext}
Assume the same setting as \cref{thm:maintext}. Suppose, additionally, there exists a sufficiently
large $C_{1, \mathcal{H}}>0$ such that $\P(\mathbf{A}_n(C_{1, \mathcal{H}})) \ge 1-
\frac{1}{n^2}$. Then, there exists a constant $C_{0, \mathcal{H}}$ such that the expected
regret is dominated by the rate function \[
\E\bk{
    \mathrm{MSERegret}_n(\hat G_n, \hat\eta) } \le C_{0, \mathcal{H}} n^{-\frac{2p}{2p+1}} (\log n)^{\frac{2+\alpha}{\alpha} + 3 + 2\beta_0}.
\]
\end{restatable}

\begin{rmksq}[On the additional assumption in \cref{cor:maintext}]
The additional assumption required by \cref{cor:maintext} is implied by exponential-tail
behavior of $\max\pr{\norm{\hat s - s_0}_\infty , \norm{\hat s - s_0}_\infty}$. Suppose
that, for some $m, c_1, c_2, t_0 > 0$, for all $ t > t_0$ \[
\P\pr{\max\pr{\norm{\hat s - s_0}_\infty , \norm{\hat m - m_0}_\infty} > t n^{-\frac{p}{2p+1}} (\log n)^{\beta_1}} \le c_1 e^{-c_2 t^m}.
\]
Then setting $t = C (\log n)^{1/m}$ and $\beta_0 = \beta_1 + 1/m$ shows that  $\P(\mathbf{A}_n)
\le \frac{1}{n^2}$ for some appropriate choice of $C$. This is in turn implied by
exponential concentration of the entrywise error: There exists $c_3, c_4, m, t_0 > 0$ such
that for all $t > t_0$, \[ \text{For all $i = 1,\ldots, n$}:
\P\pr{\max\pr{|\hat s(\sigma_i) - s_0(\sigma_i)|, |\hat m(\sigma_i) - m_0(\sigma_i)|} > t
n^{-\frac{p}{2p+1}} (\log n)^{\beta_2}} \le c_3 e^{-c_4 t^m},
\]
via a union bound. In essence, the additional assumption strengthens the Markov-type
control of $\P(\mathbf{A}_n)$ that is automatically available via \cref{as:holder}.
\end{rmksq}

It is easy to show a lower bound on the regret that matches the polynomial part of the
rate: We do so by showing that a good posterior estimate $\hat\theta_i$ implies a good
estimate $\hat m(\sigma_i)$, which then allows us to apply standard minimax lower bound
results for nonparametric regression. Note that since we treat $\sigma_{1:n}^2$ as fixed,
we state the lower bound for some specific instances of $\sigma_{1:n}^2$.

\begin{restatable}{theorem}{thmminimaxlower}
\label{thm:minimaxlower}

There exists instances of $(\sigma_{1:n}, A_1)$ satisfying \cref{as:variance_bounds} such
that, for some constant $c_\mathcal{H}$, \[
\inf_{\hat\theta_{1:n}} \sup_{\eta_0, G_0} \E_{\eta_0, G_0} \bk{
    \frac{1}{n} \sum_{i=1}^n (\hat\theta_{i} - \theta_i)^2 - (\theta_{i}^* - \theta_i)^2
} \ge c_\mathcal{H} n^{-\frac{2p}{2p+1}}
\]
where the infimum is taken over all (possibly randomized) estimators of $\theta_{1:n}$,
and the supremum is taken over all $G_0, \eta_0$ satisfying
\cref{as:holder,as:moments,as:variance_bounds}.
\end{restatable}

As a result, the rate \eqref{eq:regret_rate_final} is optimal, up to logarithmic factors.
We may view the additional logarithmic factors as the price of having to estimate $G_0$
via NPMLE, but this cost is not substantial.

\begin{rmksq}[Technical discussion of \cref{cor:maintext,cor:maintext,thm:minimaxlower}]
  \begin{enumerate}[wide]
  \item \Cref{thm:maintext,cor:maintext} are finite-sample statements. As a result, they
  hold uniformly over the problem parameters $\mathcal{H}$ that feature in
  \cref{as:holder,as:moments,as:variance_bounds}. However, the usefulness of
  \cref{thm:maintext,cor:maintext} still lies in the convergence rate, as the constant
  implied by the proofs is likely larger than optimal.

  \item These results should readily extend to the case where covariates are present and
  the location-scale assumption is with respect to the additional covariates $X_i$:
  $\theta_i \mid \sigma_i \sim G_0\pr{\frac{\theta_i - m_0(X_i, \sigma_i)}{s_0(X_i,
  \sigma_i)}}$, under assumption on the estimator $\hat m, \hat s$ analogous to
  \cref{as:holder}. Of course, the resulting convergence rate would suffer from the curse
  of dimensionality, and the term $n^{-\frac{2p}{2p+1}}$ would be replaced with $n^{-
  \frac{2p}{2p+1+d}}$, where $d$ is the dimension of $X$.

  These results should also readily extend to accommodate any estimator of $\eta_0$---not
  necessarily those that satisfy \cref{as:holder}---if we allow for sample-splitting. At a
  high level, assume $\Delta_n$ is a rate function where $\P(
  \norm{\hat\eta - \eta_0}_\infty \le
  C \Delta_n) \to 1$. Suppose we divide the data into $K$ folds, where $\hat\eta_k$ is
  estimated on the other ($[K]\setminus\br{k}$) folds and evaluated on the $k$\th{} fold:
  $\hat\eta_i = \hat\eta_k(\sigma_i)$ for observation $i$ in fold $k$. Then the resulting
  convergence rate analogous to \cref{thm:maintext} is $\Delta_n^2 (\log n)^{\frac{2+\alpha}
  {\alpha} + 3}.$

  \item The lower bound in \cref{thm:minimaxlower} is proved with known $s_0$ and known
  $G_0 \sim \Norm(0,1)$, and reduces to the minimax lower bound for learning $m_0$. Thus,
  having to estimate $G_0$ with NPMLE only worsens the regret rate by logarithmic factors.

  \item \Cref{thm:maintext,cor:maintext} are proved in \cref{sub:regret_decomp} by a
  direct application of \cref{thm:regret_on_An} in \cref{sec:bayesrisk}. The main ideas
  from the proof are discussed in more depth in \cref{asub:proofideas}. \Cref
  {thm:minimaxlower} is proved in \cref{sec:minimax}.

  The proof modifies and extends the argument in \cite{jiang2020general} and
  \cite{soloff2021multivariate}. Substantial extension is necessary due to $\eta_0$ being
  unknown---since the NPMLE $\hat G_n$ is over the estimated data $\hat Z_i, \hat\nu_i$
  and not the unobserved data $Z_i, \nu_i$. To do so, two key ingredients are an oracle
  inequality for the likelihood (\cref{cor:suboptimality} in \cref{sec:likelihood}) and a
  large-deviation inequality for the average squared Hellinger distance of the marginal
  distribution of the data (\cref{cor:hellinger_large_dev} in \cref{sec:hellinger}). These
  results may be of independent interest.

  \item Relative to the literature \citep{jiang2020general,soloff2021multivariate}, we
  only consider the case where $G_0$ is thin-tailed (\cref{as:moments}). Relative to
    \cite{soloff2021multivariate}, we only consider the case where $Y_i, \theta_i$ are
    both scalars.

\end{enumerate}
\end{rmksq}

\begin{rmksq}[Robustness to \eqref{eq:location_scale}]
Our result shows that the nonparametric EB methods we propose is optimal under the
location-scale assumption \eqref{eq:location_scale}, in the sense that they approximate
the oracle Bayesian's Bayes risk at near-optimal rates. Such an optimality result does
rely on \eqref{eq:location_scale} being a plausible description of the data-generating
process. Here, we make a few informal comments regarding \eqref{eq:location_scale} and
potential misspecification.

\begin{enumerate}[wide]
  \item Suppose \eqref{eq:location_scale} is misspecified, and let $G_{(i)}$ denote the
  distribution of $\theta_i \mid \sigma_i$. Our estimators for $\hat G_n$ and $\hat\eta$
  do nevertheless imply an estimated prior $\hat G_{(i)}$, and we use the posterior mean
  under $\hat G_{(i)}$ for predicting $\theta_i$.

  A result by \cite{jiang2009general}, reproduced as \cref{lemma:misspec_prior},
   relates the regret of $g$-modeling with a misspecified prior with the average
  squared Hellinger distance of the marginal distribution of the data. Informally, \[
\frac{1}{n} \sum_{i=1}^n \int \pr{\hat\theta_{i, \hat G_{(i)}, \sigma_i} (y) - \theta^*_i
(y)}^2 f_{G_{(i)},
\sigma_i}(y)\,dy \lessapprox \frac{C (\log n)^3 }{n} \sum_{i=1}^n h^2\pr{f_{G_{(i)},
\sigma_i},
f_
{\hat G_{(i)}, \sigma_i}}
  \]
  where (i) $f_{G, \sigma}(y) = \int \varphi\pr{\frac{y-\theta}{\sigma}}\frac{1}{\sigma} G
  (d\theta)$ is the marginal PDF of $Y$ under prior $G$ and variance $\sigma^2$ and (ii)
  $h^2(p,q) = 1-\int_\R \sqrt{pq}$ is the squared Hellinger distance.

  Thus, as long as the estimated prior $\hat G_{(i)}$ generates marginal distributions of
  the data that is close to the true marginal distributions, the proposed procedure may
  still perform well. For that to happen with our estimation scheme, we would hope that
  the NPMLE $\hat G_n$ approximates some common features of $G_{(i)}$, so that the $\hat
  G_{(i)}$ transformed from $(\hat G_n, \hat\eta)$ is close to $G_{(i)}$.
  Even when \eqref{eq:location_scale} does not exactly hold,
  intuitively, demeaning and normalizing by the conditional moments
  \eqref{eq:transformed_data} still brings $G_{(i)}$ closer to each other, and there is
  greater hope that $\hat G_n$ estimated on the transformed data
  \eqref{eq:transformed_data} captures of the common features of $G_{(i)}$.

  \item There are, of course, other ways of bringing the conditional distributions $G_{
  (i)}$ closer. We think matching the first two moments is a particularly convenient and
  attractive strategy. Under exact normality $Y_i \mid \theta_i, \sigma_i^2 \sim \Norm
  (\theta_i, \sigma_i^2)$, if we commit to a strategy where we perform $g$-modeling on
  some transformed data, only affine transformations preserve the conditional normality of
  $Y_i$. This constraint limits us to two free parameters for each $Y_i$, and it seems
  natural for those to be the first and second moment of the conditional distribution.

  Indeed, Theorem 2.5 in \cite{chen2022asymptotics} shows that, as $\sigma \to \infty$, if
  $G_1$ and $G_2$ match the first $k$ moments, then, for some quantity $C_k(G_1, G_2)$
  independent of $\sigma$, \[
 \KL{f_{G_1, \sigma}}{f_{G_2, \sigma}} = \frac{C_k(G_1, G_2)}{\sigma^{k+1}} + o\pr{
 \frac{1}{\sigma^{k+1}}}.
  \]
  Thus, at least for large $\sigma$, the KL-divergence\footnote{As a reminder, KL
  divergence dominates Hellinger distance by Pinsker's inequality.} between $f_{G_1,
  \sigma}$ and $f_ {G_2,  \sigma}$ depends on the number of
  matched moments to a first order. In this sense, standardizing the first two moments
  \eqref{eq:transformed_data} is a very natural approach to bring the conditional
  distributions $G_{(i)}$ closer.
\end{enumerate}
\end{rmksq}

\subsection{Other methods for dealing with non-independence}
\label{rmksq:nonindep}

At its core, the problem with heteroskedasticity is that the distribution of the data
$Y_i$ depends on a parameter $\theta_i$ and a nuisance parameter $\sigma_i$, but the
nuisance parameter is \emph{known}, and may not be independent from $\theta_i$. Hence, an
oracle Bayesian who knows $\theta_i \mid \sigma_i$ may be extremely powerful relative to
the empirical Bayesian, who has a hard time estimating $\theta_i \mid \sigma_i$ without
further assumptions. Our proposed solution to this problem is to assume that the
distribution of $(\theta_i,
\sigma_i)$ is well-approximated by \eqref{eq:location_scale}. We thus rely on the
resulting independence of $\tau_i$ and $\sigma_i$.

There are alternative solutions available, though, like ours, they impose some
restrictions beyond the heteroskedastic Gaussian location model. In this subsection we
introduce some of them, and discuss these restrictions.

\subsubsection{Treat the nuisance parameter as unknown}
One potential solution is to treat $S_i \equiv \sqrt{n_i}\sigma_i$ as an estimator for
some $\sigma_ {i0}$, and rely on the fact that $(Y_i, S_i^2)$ has a known distribution
\citep{banerjee2020nonparametric,gu2017empirical}. However, doing so still relies on the
independence between some parameter and the some nuisance parameter:\footnote{Of course,
this type of independence may be more palatable than $\sigma_i \indep \theta_i$.}

\begin{enumerate}[wide]
  \item If the analyst has access to microdata,
  \cite{banerjee2020nonparametric} and \cite{gu2017empirical} propose the following
  parametric model:
  \begin{align*}
  Y_{ij} \mid \sigma_{i0}, \theta_i, n_i &\iid \Norm(\theta_i, \sigma_{i0}^2) \quad j =
1,\ldots,n_i. \numberthis \label{eq:micro_normality}
\end{align*}
Under the normality assumption on the microdata, we can define the sample moments
\begin{align*}
Y_i \equiv \frac{1}{n_i} \sum_{j=1}^n Y_{ij} \mid \sigma_{i0}, \theta_i, n_i &\sim \Norm
\pr{\theta_i, \frac{\sigma_{i0}^2}{n_i}
}
\\
S_{i}^2 \equiv \frac{1}{n_i - 1} \sum_{j=1}^n (Y_{ij} - Y_i)^2 \mid \sigma_{i0}, \theta_i, n_i &\sim
\Gammadist\pr{
  \frac{n_i - 1}{2}, \frac{1}{2\sigma_{i0}^2}
}\tag{$\Gammadist(\alpha, \lambda)$ is the (shape, rate) parametrization} \\
Y_i &\indep S_i^2 \mid \theta_i, \sigma_{i0}, n_i
  \end{align*}
Let $\mathbf{Y}_i = (Y_i, S_i^2)$ collect the sample mean and variance, and let $\bm{\theta}_i =
(\theta_i, \sigma_{i0}^2)$ collect the mean parameter and the variance parameter, then
$\mathbf{Y}_i \mid \bm{\theta}_i, n_i \sim \bm{f}(\cdot \mid \bm{\theta}_i, n_i)$  has a known, bivariate
exponential family likelihood. \emph{Under the assumption that $\bm{\theta}_i \indep n_i$}, we
can estimate the distribution $\bm{\theta}_i \sim \mathbf{G}$ (as in the $g$-modeling approach in
\citet{gu2017empirical}), or the distribution $\mathbf{Y}_i \sim \bm{f}_{\mathbf{G}}$ and use Tweedie's
formula (as in the $f$-modeling approach in \citet{banerjee2020nonparametric}). However,
both approaches do still require that the redefined parameter $\bm{\theta}_i$ is independent
from a nuisance parameter $n_i$, which also enters the likelihood $\mathbf{Y}_i \mid \bm{\theta}_i,
n_i$.
\item In general, this problematic dependence is not solely with the sample size $n_i$.
Without the normality assumption \eqref{eq:micro_normality}, one could use the
asymptotic approximation \[
\colvecb{2}{\sqrt{n_i} (Y_i - \theta_i)}{ \sqrt{n_i} (S_i^2 - \sigma_{i0}^2)}
\underset{n_i\to\infty}{\dto} \Norm(0, \Sigma_i)
\]
and treat $\mathbf{Y}_i \mid \bm{\theta}_i, \Sigma_i, n_i \sim \Norm\pr{\bm{\theta}_i, \frac{\Sigma_i}
{n_i}}$ for known $(\mathbf{Y}_i, \Sigma_i/n_i, n_i)$. However, in general, the off-diagonal and
lower-right entries of $\Sigma_i$ depend on higher moments of $Y_{ij}$ and are not
functions of $\bm{\theta}_i$.
\end{enumerate}

\subsubsection{Normalize the nuisance parameter away}
 Another potential solution is to normalize the nuisance parameter away, by working with
$t$-statistics: \[
T_i \equiv \frac{Y_i}{\sigma_i} \mid (\sigma_i, \theta_i) \sim \Norm\pr{\frac{\theta_i}
{\sigma_i},1},
\]
as the likelihood for $t$-statistics is homoskedastic by construction. Naturally, we
consider performing empirical Bayes analysis assuming that $\frac{\theta_i}{\sigma_i} \iid
H_0$, and use, say, $\sigma_i \mathbf{E}_{\hat H_n, 1}\bk{\frac{\theta_i}{\sigma_i} \mid T_i}$ as
an estimator for the posterior mean of $\theta_i$. However, it is unclear how to relate
such an estimator to an optimal estimator for an \emph{unweighted objective}, even in some
restricted class of decision
rules.
\begin{enumerate}[resume, wide]
  \item Consider decision rules of the form $\delta_{\text{t-stat}}(Y, \sigma) = \sigma h
  (Y/\sigma)$.
  Under the compound loss \[
\B_n(\{\delta_{\text{t-stat}}\}, \sigma_{1:n}^2) = \E\bk{\frac{1}{n} \sum_{i=1}^n (\sigma_i h(T_i) -
\theta_i)^2} = \E\bk{\frac{1}{n} \sum_{i=1}^n \sigma^2_i \pr{h(T_i) -
\frac{\theta_i}{\sigma_i}}^2},
  \]
  we readily find that the optimal decision rule---in this restricted class---is given by
  \[
h^\star(T_i) =\frac{\E\bk{\sigma_i \theta_i \mid T_i}}{ \E[\sigma_i^2 \mid T_i]},
  \]
  which solves a weighted least squares problem. Note that the optimal $h$ is
  not the posterior mean of $\theta_i/\sigma_i$ given $T_i$, unless $\sigma_i^2 \indep
  \frac{\theta_i}{\sigma_i}$.
  \item However, if we have a weighted compound loss objective, then $\sigma_i \E[
  \theta_i/\sigma_i \mid
  T_i]$ is indeed optimal within the class of $\delta_{\text{t-stat}}$, since, in this
  case, the weighted
compound loss is equal to the compound loss on the $t$-statistic space:
\[\B_n^{\text{weight}}(\{\delta_{
\text{t-stat}}\}, \sigma_{1:n}^2) = \E\bk{\frac{1}{n} \sum_{i=1}^n \frac{1}
{\sigma_i^2} (\sigma_i h(T_i) -
\theta_i)^2} = \E\bk{\frac{1}{n} \sum_{i=1}^n \pr{h(T_i) -
\frac{\theta_i}{\sigma_i}}^2}.
  \]
  Nevertheless, we emphasize that the optimality is with respect to a restricted class of
  decision rules.
\end{enumerate}

\subsubsection{Variance-stabilizing transformation} A leading case where $\theta_i
\not\indep
\sigma_i$ is when the underlying microdata are Bernoulli. Specifically, we rely on the
asymptotic approximation \[
\sqrt{n_i} (Y_i - \theta_i)\underset{n_i\to\infty}{\dto} \Norm(0, \theta_i (1-\theta_i)).
\]
A variance-stabilizing transform can disentangle the dependence: Let
$W_i = 2\arcsin(\sqrt{Y_i})$ and $\omega_i = 2\arcsin(\sqrt{\theta_i})$, and, by the delta
method, \[
\sqrt{n_i} \pr{W_i - \omega_i} \underset{n_i\to\infty}{\dto} \Norm(0, 1). \quad \text{
Thus, approximately, } W_i \mid \omega_i, n_i \sim \Norm\pr{\omega_i, \frac{1}{n_i}}.
\]
We remark that
\begin{enumerate}[resume, wide]
  \item $W_i$ is still heteroskedastic, and the transformed parameter $\omega_i$ may
  still correlate with the nuisance parameter $n_i$.
  \item To optimize compound squared-error loss on $\theta_i$, we would estimate the
  posterior mean $\E[\sin^2(\omega_i/2) \mid W_i]$. To the best of our knowledge, a regret
  rate result like \cref{thm:maintext} is not known for the posterior mean of transformed
  parameters.\footnote{In particular, since Tweedie's formula does not apply to $\E[\sin^2
  (\omega_i/2) \mid W_i]$, it is unclear how to relate this quantity to the marginal
  distribution of $W_i$. }
\end{enumerate}

\subsection{Other decision objectives and relation to squared-error loss}

So far, our results concern mean-squared error loss. In this subsection, we show that
regrets in Bayes risk for two different decision objectives are dominated by the regret in
the Bayes risk for squared error loss.

\begin{enumerate}
  \item \textsc{utility maximization by selection}: Assume that for each unit $i$, the
  decision-maker makes a binary decision $a_i \in \br{0,1}$, such that their payoff is
  $a_i (\theta_i - c_i)$, for known costs $c_1,\ldots, c_n \in \R$. The average payoff is
  \[U_n(a, \theta) = \frac{1}{n} \sum_{i=1}^n a_i(\theta_i - c_i).
  \]
  \item \textsc{top-$m$ selection}{} \citep{coeyhung}: For a fixed $1 < m \le n$, the decision-maker
  selects $m$ entities, i.e., they make binary decisions $a_1,\ldots, a_n \in \{0,1\}^{n}$
  such that $\sum_{i=1}^n a_i = m$. The payoff is proportional to the average $\theta_i$
  of the selected set: \[
U_n(a,\theta) = \frac{1}{n} \sum_{i=1}^n a_i \theta_i.
  \]
\end{enumerate}

Note that, under $P$, the oracle Bayesian's decision rule takes the form of ranking by
posterior mean:\footnote{The posterior risk for any decision rules $\delta_i$ measurable
with respect to the data is $
\frac{1}{n} \sum_{i=1}^n \delta_i (\E[\theta_i \mid Y_i, \sigma_i] - c_i)
$ for \textsc{utility maximization by selection}, and $\frac{1}{n} \sum_{i=1}^n \delta_i \E[\theta_i
\mid Y_i, \sigma_i]$ for \textsc{top-$m$ selection}.}
\begin{enumerate}
  \item \textsc{utility maximization by selection}: $\delta_i^\star(Y_{1:n} ; \sigma_{1:n}) =
  \one\pr{\E
  [\theta_i
  \mid Y_i, \sigma_i] \ge c_i}$.
  \item \textsc{top-$m$ selection}: $\delta_i^\star(Y_{1:n} ; \sigma_{1:n}) =
  \one\pr{\E
  [\theta_i
  \mid Y_i, \sigma_i] \text{ is among the largest-$m$ posterior means}}$.
\end{enumerate}
Let $\delta^\star$ be the oracle Bayesian's decision rule in either case. Let
$\hat\theta_1,\ldots,\hat\theta_n$ be a vector of estimated posterior means,\footnote{We
allow for any measurable function of the data $(Y_{i}, \sigma_i)_{i=1}^n, U$, where $U
\sim \Unif[0,1]$ is an independent random noise. The inclusion of $U$ incorporates
randomized estimators. } and let $\hat\delta$ denote the analogous decision rules, using
 $\hat\theta_{1:n}$ instead of $\theta^*_{1:n}$. Define $\mathrm{UMRegret}_n$ and $\mathrm{TopRegret}_{n}^{(m)}$ as
 the difference \[ U_n(\delta^\star, \theta) - U_n(\hat\delta, \theta).
\]
The following result relates $\mathrm{UMRegret}_n$ and $\mathrm{TopRegret}_{n}^{(m)}$ to $\mathrm{MSERegret}_n$.

\begin{restatable}{theorem}{mserelevance}
\label{thm:mserelevance}

We have \begin{enumerate}
  \item For \textsc{utility maximization by selection}, \[
\E[\mathrm{UMRegret}_n] \le \E\bk{\frac{1}{n} \sum_{i=1}^n (\hat\theta_i - \theta_i^*)^2}^{1/2} .
  \]
  \item For \textsc{top-$m$ selection}, \[
\E[\mathrm{TopRegret}_{n}^{(m)}] \le 2\sqrt{\frac{m}{n}} \E\bk{\frac{1}{n} \sum_{i=1}^n (\hat\theta_i -
\theta_i^*)^2}^{1/2}.
  \]
\end{enumerate}
\end{restatable}

\begin{rmksq}
\begin{enumerate}[wide]
  \item \Cref{thm:mserelevance} shows that the two decision problems \textsc{utility maximization by selection}{} and
  \textsc{top-$m$ selection}{} are easier than estimating the oracle Bayesian posterior means. As a result, our
  convergence rates from \cref{thm:maintext,cor:maintext} also upper bound regret rates
  for these two decision problems.

  \item \textsc{top-$m$ selection}{} is studied by \cite{coeyhung}, who show that in the $\theta \indep
  \sigma_i^2$ setting,
  if $\hat\theta_{1:n}$ are $g$-modeling posterior means, i.e., posterior means for some
  prior $\hat G$ of $G_{(0)}$, then \[
\E[\mathrm{TopRegret}_{n}^{(m)}] = O\pr{d^2(G_{(0)}, \hat G)}.
  \]
  In nonparametric settings, $G_{(0)}$ is often only estimable at logarithmic rates, and
  thus the rate in \cref{thm:mserelevance} is more favorable in those settings.
  \Cref{thm:mserelevance} is worse than the rate in \cite{coeyhung} for parametric
  settings, suggesting that our bound is not tight.

  \item The bound for \textsc{utility maximization by selection}{} actually upper bounds $\mathrm{UMRegret}_n$ with the
  $\ell_1$-distance between $\hat\theta$ and $\theta^*$. However, since the $L_1$ and
  $L_2$ minimax rates for nonparametric regressions are the same
  \citep{tsybakov2008introduction}, upper bounding the $\ell_1$ distance with the $\ell_2$
  distance is not loose in the worst case.

  \item We can think of \textsc{top-$m$ selection}{} as a decision problem for recommending opportunity areas
  for potential movers \citep{bergman2019creating}. Specifically, suppose we are tasked
  with selection one-third of Census tracts in a certain area and recommending individuals
  to move to these Census tracts. We would like to choose these Census tracts so as to
  maximize some notion of average economic mobility. \textsc{top-$m$ selection}{} takes the view that movers
  may move with equal probability to these Census tracts.

  We prove in \cref{asec:mserelevance} the bound (2) by relating the regret to the sorted
  difference in the two vectors of posterior means. We then bound the sorted difference in
  terms of the unsorted difference by the rearrangement inequality. Our proof can be
  slightly generalized to the decision objective where $ U_n(a, \theta) = \frac{1}{n}
  \sum_{i=1}^n a_i \theta_i
  $
  for $(a_1, \ldots, a_n)$ being a permutation of some given  weights $w_1, \ldots, w_n >
  0$. \textsc{top-$m$ selection}{} is the special case where $w_i \in \{0,1\}$ and $\sum_i w_i = m$.

  Another special case of this generalized problem is to rank Census tracts according to
  economic mobility for people to move to, but operate under the assumption that the
  probability they move to a particular place depends decreasingly and solely on its rank.
  For this problem, the oracle Bayesian's decision rule is to rank their posterior means
  $\theta_i^*$ and assign the largest $w_i$ to the largest $\theta_i^*$, the second
  largest to the second largest, and so forth. We obtain a bound of \[
  2\sqrt{\frac{1}{n}\sum_{i=1}^n w_i^2} \cdot \E\bk{\frac{1}{n} \sum_{i=1}^n (\hat\theta_i
  -
\theta_i^*)^2}^{1/2}.
  \]
  with the same argument.

\end{enumerate}
\end{rmksq}

\section{Empirical illustration}
\label{sec:empirical}

The rest of this note concludes with an empirical application calibrated to the
Opportunity Atlas data in \cite{chetty2018opportunity}, simulating a policy exercise
resembling Creating Moves to Opportunity \citep{bergman2019creating}. We first briefly
summarize
the approaches taken in each of these papers.

\subsection{The Opportunity Atlas and Creating Moves to Opportunity}

For each Census tract $i$, letting $y_{k}$ denote the (individual-level) outcome of a
randomly sampled individual $k$ from the United States, \cite{chetty2018opportunity}
estimates \[
\tilde \theta_i(\mathrm{g}, \mathrm{r}, \mathrm{fr}) = \E[y_{k} \mid \text{Tract}_k = i,
\text{Gender}_k=
\mathrm{g}, \text{Race}_k=\mathrm{r}, \text{FamilyIncomeRank}_k=\mathrm{fr}].
\]
with a regression specification of the following form  within a particular race-gender
cell: For $i(k)$ denoting the tract of individual $k$, \[ y_{k} = \alpha_{i(k)} +
\beta_{i(k)} \hat f(\text{FamilyIncomeRank}_k) + U_k
\]
where $\hat f$ is a nonparametric estimate of $\E[y_{k} \mid \text{Gender}_k=
\mathrm{g}, \text{Race}_k=\mathrm{r}, \text{FamilyIncomeRank}_k]$,
pooling over all Census tracts. In short, this specification limits the heterogeneity of
how the outcome depends on family income rank by Census tract, in exchange for greater
precision. For binary outcomes, the regression estimates do not respect the $
[0,1]$-support of predicted values.

Taking the estimates from the Opportunity Atlas, \cite{bergman2019creating} conduct an
experiment in King County, Washington (Seattle). They provided assistance to treated
low-income individuals\footnote{They are families with a child below age 15 who are issued
Section 8 vouchers between April 2018 and April 2019, with median household income of
\$19,000. About half of the sampled households are Black and about a quarter are white
\citep[Table 1,][]{bergman2019creating}.} to move to ``Opportunity Areas''---Census tracts
with favorable estimates of economic mobility. Specifically, the Opportunity Areas are
Census tracts within King County with posterior means for $\tilde
\theta_i$ ranking among the top third of Census tracts, according to the empirical Bayes
procedure used by the authors. The authors find that their recommendations did cause
households in the study to move to areas with higher predicted economic mobility.

Specifically, let $(\tilde Y_i, \sigma_i)$ denote the point estimate and standard error
from the Opportunity Atlas for Census tract $i$. \cite{bergman2019creating} use the
following empirical Bayes procedure:\footnote{We abstract away from certain practical
choices made by the authors, such as making sure that the selected Census tracts are
contiguous. See \cite{bergman2019creating} for details. } \begin{enumerate}[wide]
  \item Residualize $\tilde Y_i$ on a list of covariates $X_i$. Let the fitted values be
  $\hat y_i$ and the residuals be $Y_i = \tilde Y_i - \hat y_i$. Define $\theta_i = \tilde
  \theta_i - \hat y_i$. Treat $Y_i \mid \theta_i,
  \sigma_i \sim \Norm(\theta_i, \sigma_i^2)$.
  \item  Assume $\theta_i \mid \sigma_{1:n}^2 \sim \Norm
  (m_0, s_0^2)$.
  \item Estimate $m_0$ with $\hat m$ and $s_0^2$ with $\hat s^2$, and output the posterior
  mean estimates for $\tilde \theta_i$ by adding back the fitted values $\hat y_i$ \[
\hat{\tilde \theta}_i^{(\textsf{IndepGauss})} = \hat y_i + \hat m + \frac{\hat s^2}{ \hat s^2 + \sigma_i^2 } (Y_i -
\hat m).
  \]
  Here, $ \hat m + \frac{\hat s^2}{ \hat s^2 + \sigma_i^2 } (Y_i -
\hat m)$ is the posterior mean of $\theta_i$ given $Y_i$ under the parametric assumptions
in the previous step (2), with $m_0$ replaced by $\hat m$ and $s_0$ replaced by $\hat s$.
\end{enumerate}

We shall refer to this strategy as \textsf{IndepGauss}, and it shall be the main benchmark that
we compare against. In contrast, we propose the following steps to replace (2) and (3),
which we spell out as a reminder.
Our first strategy, which we call \textsf{PreferredGauss}{} is
\begin{enumerate}[label=(\textsf{PreferredGauss}-\arabic*),itemindent=5em,wide]
  \setcounter{enumi}{1}
  \item Assume $\theta_i \mid \sigma_i \sim \Norm(m_0(\sigma_i), s_0^2(\sigma_i))$.
  \item Estimate $m_0$ with $\hat m$ and $s_0^2$ with $\hat s^2$, and output the posterior
  mean estimates for $\tilde \theta_i$ by adding back the fitted values $\hat y_i$ \[
\hat{\tilde \theta}_{i}^{(\textsf{PreferredGauss})} = \hat y_i + \hat m(\sigma_i) + \frac{\hat s^2
(\sigma_i)}{\hat s^2(\sigma_i) +
\sigma_i^2 } (Y_i -
\hat m (\sigma_i)).
  \]
\end{enumerate}

Our second strategy, which we call \textsf{PreferredNP}{} is
\begin{enumerate}[label=(\textsf{PreferredNP}-\arabic*),itemindent=5em,wide]
  \setcounter{enumi}{1}
  \item Assume $\theta_i \mid \sigma_i \sim G_0(m_0(\sigma_i), s_0^2(\sigma_i))$.
  \item Estimate $m_0$ with $\hat m$ and $s_0^2$ with $\hat s^2$. Estimate $G_0$ as in
  \eqref{eq:npmle}. Output the posterior
  mean estimates for $\tilde \theta_i$ by adding back the fitted values $\hat y_i$ to the
  posterior mean for $\theta_i$.
\end{enumerate}

\subsection{Calibrated simulation design and competing approaches}

\newcommand{\textsf{Naive}}{\textsf{Naive}}
\newcommand{\textsf{Oracle}}{\textsf{Oracle}}
\newcommand{\textsf{MVOracle}}{\textsf{MVOracle}}
\newcommand{\textsf{IndepNPMLE}}{\textsf{IndepNPMLE}}

Let $(\tilde Y_i, \sigma_i)$ be the estimate and standard error from the Opportunity
Atlas---consisting of all 10,109 tracts within the twenty largest CZs---for some parameter
$\tilde \theta_i$ at the tract level. We now describe a data-generating process for
simulated parameter $\tilde \theta_i^*$ and simulated data $\tilde Y_i^*$:
\begin{enumerate}[wide]
  \item We residualize $\tilde Y_i$ by some covariates $X_i$  and obtain the fitted values $\hat y_i$ and residuals
  $Y_i$.\footnote{The covariates used are listed in the caption for
  \cref{fig:kfr_pooled_raw_data}.}
  \item We estimate $\hat m_0(\sigma) = \E[Y \mid \sigma]$ and $\hat s_0(\sigma) = \var(Y
  \mid \sigma) - \sigma^2$ via $k$-nearest neighbors on the data $(Y_i, \log
  (\sigma_i))$.\footnote{The raw $k$-NN estimates for $s_0$ might be negative. We truncate
  those estimates at zero. Furthermore, we set $\hat s_0$ to be 0.8 times the raw
  estimates and 0.2 times the sample mean of the raw estimates, all scaled by 1.1, which
  is always strictly positive and slightly larger than the true $s_0$. }
  \item Define \[r_i = \Xi\pr{\frac{Y_i - \hat m_0(\sigma_i)}{\sqrt{\sigma_i^2 +
  \hat s_0^2}}}
  \minwith 1\] for some function $\Xi$.\footnote{In these simulations, $
  \Xi(t) = \frac{1}{1.1}|t|^{1/10}.
  $ This has the effect of pushing the masses $r_i$ to be closer to 1, and pronouncing
  the bimodality of the conditional distribution $\tau_i \mid \sigma_i$. Other choices of
  $\Xi$ generate broadly similar patterns, except that \textsf{Oracle}{} and \textsf{MVOracle}{} (defined
  later) are even
  more similar. } The
  distribution for $\tau_i^* \mid \sigma_i$ is
  the two-point
  mixture of Gaussians: \[
\tilde \theta_i^* \mid \sigma_i \sim \frac{1}{2} \Norm\pr{-r_i, v} + \frac{1}{2} \Norm
(r_i, v) \quad v = 1-r_i^2. \numberthis
\label{eq:oracle_prior}
  \]
  which is mean-zero and unit variance by construction. Note that since $r_i$ may be
  different for each $\sigma_i$, this data-generating process does not exactly obey the
  location-scale model
  \eqref{eq:location_scale}.
  \item Then $\theta_i^* = \tau_i^* \hat s_0(\sigma_i) + \hat m_0(\sigma_i)$, $Y_i^*
  \mid \theta_i^*, \sigma_i \sim \Norm(\theta_i^*, \sigma_i^2)$, $\tilde Y_i^* =
  Y_i^* + \hat y_i$, and $\tilde \theta_i^* = \theta_i^* + \hat y_i$.
\end{enumerate}
The above steps enumerate how to produce one draw of $\tilde \theta_i^*, \tilde Y_i^*$.
Note that, for the simulated data, the true $m_0$ is $\hat m_0$ and the true $s_0$ is
$\hat s_0$. \Cref{fig:example_sim} shows an overlay of real and simulated data for one of
the variables we consider.

\begin{figure}[htb]
  \centering
  \includegraphics[width=\textwidth]{assets/compare_real_sim_data.png}
  \caption{A draw of real vs. simulated data for estimates of $\P(\text{Top 20 Income}
  \mid \text{Black}, \text{25\th{} percentile parent income})$}
  \label{fig:example_sim}
\end{figure}

On the simulated data $(\tilde Y_i^*, \sigma_i)$, we then put several empirical Bayes
strategies to test for a variety of policy exercises. In addition to \textsf{IndepGauss},
\textsf{PreferredGauss}, and \textsf{PreferredNP}, we also consider the following strategies:
\begin{itemize}[wide]
  \item \textsf{Naive}: Use the estimates $\tilde Y_i$ directly.
  \item \textsf{Oracle}: A Bayesian who knows \eqref{eq:oracle_prior}, $m_0$, and $s_0$ and
  outputs their posterior mean.
  \item \textsf{MVOracle}: A Bayesian who knows $m_0, s_0$ and uses the Gaussian-prior posterior
  mean, much like \textsf{PreferredGauss}. This is also the best linear-in-$Y_i$ decision rule.
  \item \textsf{IndepNPMLE}: Much like \textsf{IndepGauss}, except instead of modeling $\theta_i \mid
  \sigma_i \sim \Norm(m_0, s_0^2)$, it models $\theta_i \mid
  \sigma_i \sim G_{(0)}$ and estimates $G_{(0)}$ with NPMLE
  \citep{kiefer1956consistency,koenker2014convex,jiang2009general,jiang2020general,soloff2021multivariate}.
\end{itemize}

The outcome data we use are (along with their identifiers in the Opportunity Atlas data
in parentheses):
\begin{itemize}
  \item Mean income rank (\texttt{kfr})
  \item Probability of having income rank in the top 20 percentiles (\texttt{kfr\_top20})
  \item Incarceration rate (\texttt{jail})
\end{itemize}
possibly for different demographics cells. Since all three variables have a percent or
percentile interpretation, we scale the result by 100. We repeat each experiment 100
times; the resulting Monte Carlo variation is small, since the quantities we consider
average over 10,000 Census tracts, and have low sampling variance to start with.

\subsection{Results}

\subsubsection{Mean-squared error performance}
We first examine the performance of different empirical Bayes strategies in terms of
mean-squared error. To be precise, we consider the Bayes risk \[
\E\bk{\frac{1}{n} \sum_{i=1}^n \pr{\hat {\tilde\theta}_i - \tilde\theta_i}^2}
\]
for different posterior mean estimates $\hat {\tilde\theta}_i$. Our results are
displayed in \cref{fig:mse_table}.

We find that our preferred approaches (\textsf{PreferredNP}, \textsf{PreferredGauss}) achieve performances that
are close to the oracle approaches. \textsf{PreferredNP}{} tends to do a hair better
than \textsf{PreferredGauss}{} and sometimes better than \textsf{MVOracle}, partly due to
\eqref{eq:oracle_prior} being near-discrete. Remarkably, \eqref{eq:oracle_prior} does not
impose the location-scale model exactly, but in spite of the misspecification,
\textsf{PreferredNP}{} is able to achieve low MSE regret against the \textsf{Oracle}.

The methods that assume independence (\textsf{IndepNPMLE}, \textsf{IndepGauss}) are in the next tier of
performance, particularly for binary outcome variables and for finer demographic
categories. Their performances are similar to, though slightly worse than, that of the
preferred methods for mean income rank (which is the variable that
\cite{bergman2019creating} target). The performance gap widens for variables such as Black
male incarceration rate or the probability that a Black man having income in the top 20
percentiles. Lastly, methods that do not use covariates tend to do worse,\footnote{It may
be surprising that \textsf{IndepGauss}{} can perform worse than \textsf{Naive}{} on MSE, since Gaussian
empirical Bayes typically has a connection to the James--Stein estimator, which dominates
the MLE. We note that, as in \cite{bergman2019creating}, when we estimate the prior mean
and prior variance, we \emph{weight} the data with precision weights proportional to
$1/\sigma_i^2$. When the independence between $\theta$ and $\sigma$ holds, these precision
weights typically improve efficiency. However, the weighting does break the connection
between Gaussian empirical Bayes and James--Stein, and the resulting posterior mean does
not always dominate the MLE (i.e., \textsf{Naive}). To take an extreme example, if a particular
observation has $\sigma_i \approx 0$, then that observation is highly influential for the
prior mean estimate. If $\E[\theta_i \mid \sigma_i]$ is very different for that
observation than the other observations, then the estimated prior mean is a bad target to
shrink towards. } though \textsf{PreferredNP}{} can still leverage the information in $\theta \mid
\sigma$ and achieve good performance in certain cases.

\begin{figure}[htb]
  \centering
  \includegraphics[width=\textwidth]{assets/mse_table.pdf}
  \caption{Table of MSE Bayes risk for various empirical Bayes approaches. Each column is
  an empirical Bayes strategy that we consider, and each row is a different definition of
  $\tilde\theta_i$. All $\tilde\theta_i$'s are for individuals whose parents are at the
  25\th{} national income percentile. The first two columns show the square root of the
  squared-error Bayes risk for the \textsf{Naive}{} and \textsf{Oracle}{} strategies, scaled by 100. The
  other columns show the relative risk improvement of various methods, measured as
  \emph{the
    fraction of the gain of \textsf{Oracle}{} relative to \textsf{Naive}{} that the method captures}: $\frac{
  \text{MSE Risk of \textsf{Naive}{}}\, - \text{MSE Risk of \textsf{Method}}}{ \text{MSE Risk of
  \textsf{Naive}{}}\, - \text{ MSE Risk of \textsf{Oracle}}} \times 100$. The relative performance is
  negative if a method performs worse than \textsf{Naive}{}. The last two column with ``(no
  covs)'' are versions of the method where we do not residualize by a vector of
  covariates, and work instead with $Y_i = \tilde Y_i, \theta_i = \tilde\theta_i$.}

  \label{fig:mse_table}
\end{figure}

\subsubsection{Ranking exercise}

\afterpage{
\begin{landscape}
\begin{figure}[htb]
  \centering
  \includegraphics[width=1.4\textwidth]{assets/kfr_improv.pdf}
  \caption{Improvement over \textsf{IndepGauss}{} of selecting top $T\%$ of Census tracts according
  to posterior mean for mean income rank}
  \label{fig:kfr_improv}
\end{figure}

\begin{figure}[htb]
  \centering
  \includegraphics[width=1.4\textwidth]{assets/kfr_top20s_improv.pdf}
  \caption{Improvement over \textsf{IndepGauss}{} of selecting top $T\%$ of Census tracts according
  to posterior mean for probability of ranking in the top 20 percentiles}
  \label{fig:kfr_top20_improv}
\end{figure}

\begin{figure}[htb]
  \centering
  \includegraphics[width=1.4\textwidth]{assets/jail_improv.pdf}
  \caption{Improvement over \textsf{IndepGauss}{} of selecting top $T\%$ of Census tracts according
  to posterior mean for incarceration rate}
  \label{fig:jail_improv}
\end{figure}
\end{landscape}
}

We now consider an exercise like the decision problem \textsc{top-$m$ selection}. In particular, within each of
the CZs, we consider selecting the top $T\%$ of Census tracts according to posterior mean
estimates of the variable $\tilde \theta_i$. We assume that the decision criterion is the
average true $\tilde\theta_i$ among the tracts we select: e.g. if $\tilde\theta_i$ is mean
income rank for those at the 25\th{} parent income rank, then we would like to maximize,
on average, this variable for the selected tracts. For variables like mean income rank or
the probability that someone's income ranks in the top 20 percentiles, this exercise
selects the Census tracts with high economic mobility, perhaps so as to recommend movers
to move to these high-mobility areas, as in \cite{bergman2019creating}. Selecting Census
tracts with high incarceration rate, on the other hand, potentially informs criminal
justice reform and resource allocation decisions.

For each method, as a function of $T$, we plot the mean average $\tilde\theta_i$ among the
selected set relative to that of \textsf{IndepGauss}. That is, we plot \[
\E\bk{\frac{1}{T\% \cdot n}\sum_{i=1}^n \one\pr{\hat{\tilde\theta}_i^{(\mathsf{Method})}
\text{ is in the top $T\%$} }\tilde\theta_i - \frac{1}{T\% \cdot n}\sum_{i=1}^n
\one\pr{\hat{\tilde\theta}_i^{ (\textsf{IndepGauss})} \text{ is in the top $T\%$}
}\tilde\theta_i}.
\]
Since \textsf{IndepGauss}{} performs poorly when it does not include covariate information, we
omit that from the comparison. The results for mean income rank are collected in
\cref{fig:kfr_improv}; those for the probability of ranking in the top 20 percentiles are
collected in \cref{fig:kfr_top20_improv}; and those for the incarceration rate are
collected in \cref{fig:jail_improv}. We find that
\begin{enumerate}[wide]
\item Preferred methods generally improve over the \textsf{IndepGauss}{} benchmark and come quite
close to the performance of \textsf{Oracle}. \textsf{IndepNPMLE} and \textsf{IndepGauss} generally perform very
similarly. Among the preferred methods, \textsf{PreferredNP}{} tends to do a hair better than
\textsf{PreferredGauss}, but the difference is quite small.

\item The extent of the improvement depends on the distribution for each variable.
Generally speaking, if $\sigma_i^2$ is small (i.e., estimates are relatively precise), then
the possible room for improvement is limited, as we do not shrink very much for each
observation.
Moreover, if $\var(\E[\theta \mid \sigma])
\ll \E[\var(\theta \mid \sigma)]$---that is, if the fluctuations in $m_0(\cdot)$ is small
compared to the typical prior variance $s_0(\cdot)$---then the improvement of preferred
methods over empirical Bayes methods that assume independence is small. This is because
preferred methods exploit the heterogeneity in $m_0(\cdot)$, but if $s_0$ is large, the
prior is not informative, and so we do not shrink very much, thereby limiting the amount
of improvement.

\item
We observe small improvement for mean income rank for large demographic cells (pooled or
white). We observe moderate improvement for mean income rank for Black individuals.
Selecting the top 20\%, for instance, generates an average improvement of 0.25 percentile
ranks for Black individuals.  Roughly speaking, that corresponds to \$250 additional
income for those with income around \$19,000 as in \cite{bergman2019creating}, or an increase of
1.3\%. We note that
\textsf{Naive}{} is worse than \textsf{IndepGauss}{} by about 0.5 percentile ranks, and so moving from
\textsf{IndepGauss}{} to \textsf{PreferredNP}{} generates an improvement about half of the improvement from
using any empirical Bayes method in the first place.

\item We observe larger improvement for the binarized measure of mobility---the
probability that someone with parents from the 25\th{} income percentile has household
income ranking in the top 20\%. Roughly speaking, since the 80\th{} percentile has annual
household income of \$95,015 \citep[2015 dollars,][] {chetty2018opportunity}, one
percentage point improvement of moving to the that percentile from an annual income of
\$19,000 corresponds to an expected improvement of \$760 in dollar terms, or 4\% of
\$19,000. For Black (resp. white) individuals, the improvement from using \textsf{PreferredNP}{}
relative to
\textsf{IndepGauss}{} is about 1 percentage point (resp. 0.25--0.5) when selecting the top 20\% of
tracts, and about 2.5 percentage points (resp. 0.25--0.75) when selecting the top 10\% of
tracts within each CZ.

\item The improvements for the incarceration rate, particularly for Black men, is even
more pronounced. Selecting the top 20\% of Census tracts (resp. 10\%), we can generate an
improvement of 2 percentage points (resp. 3) over \textsf{IndepGauss}{} by using \textsf{PreferredNP}{}.
The mean tract-level incarceration rate is about 10\% for Black men for tracts in the
sample, and so these are sizable improvements.

Interestingly, the dependence between $\theta_i$ and $\sigma_i$ is so strong for the Black
male incarceration rate that
\textsf{IndepGauss}{} \emph{underperforms} \textsf{Naive}{}. As displayed in
\cref{fig:jail_black_raw_data}, higher incarceration rate correlates with higher
$\sigma_i$, partly due to the inherent relationship between Bernoulli mean and variance.
However, for any empirical Bayes strategy, observations with higher $\sigma_i$ are shrunk
more. If all observations are shrunk towards the grand mean---as in \textsf{IndepGauss}{}---the
posterior means of these high-$\sigma_i$-but-high-$\theta_i$ observations would be too
low. If the goal is to select entries with high $\theta_i$, shrinkage assuming
independence can therefore actually hurt performance.
\end{enumerate}

\subsubsection{Conclusion} To summarize, in this note, we propose empirical Bayes methods
that account for possible non-independence of $\theta_i$ and $\sigma_i$. We propose a
location-scale assumption \eqref{eq:location_scale} that leads to a natural two-step
procedure for nonparametric empirical Bayes decision rules. Under such an assumption, we
show that the resulting nonparametric empirical Bayes posterior means achieve low regret
in terms of squared error loss, with respect to the true Bayes posterior. Specifically, we
derive finite-sample upper bounds on the regret convergence rate, and show that it matches
a lower bound up to logarithmic factors. We also connect two decision problems to
estimation under squared error loss, and show that our MSE regret rates imply regret rates
for these decisions problems. Lastly, we conduct a simulation exercise calibrated to the
Opportunity Atlas \citep{chetty2018opportunity}. In simulation, the methods we propose
indeed perform well and perform only slightly worse than oracle estimators. For certain
variables in the Opportunity Atlas, using the methods we propose improve substantially
relative to using empirical Bayes methods that assume independence between $\theta_i$ and
$\sigma_i$.

\newpage

\bibliographystyle{aer}

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text:
Home Canada-CARICOM Relations: The Tipping Point
CCentral America & the Caribbean
Canada-CARICOM Relations: The Tipping Point
byAurelia Bruce
HISTORY OF THE NEGOTIATIONS
In 2008, heads of government party to the Canada-CARICOM agreement decided to move ahead with a free trade agreement. The negotiations began on November 10th 2009, and thus far, six rounds of negotiations have been held. The last round was held in March 2014, and nothing is considered agreed until the negotiations are concluded and everything is agreed on. Negotiations have taken place on market access for goods, rules of origin, trade in services and investment, financial services, intellectual property, customs procedures, dispute settlement, institutional issues, environmental and labour issues which indicates that the agreement is broad in scope.
The recent trade regime between the parties has been underpinned by the CARICOM-Canada Trade and Economic Co-operation Agreement (1979) (and its Protocols including the CARICOM-Canada 1998 Protocol on Rum); the Canada-Barbados Bilateral Investment Treaty; the Canada-Trinidad and Tobago Bilateral Investment Treaty and CARIBCAN Agreement. CARIBCAN is a non-reciprocal preferential trade agreement that grants unilateral, duty-free access to eligible goods only up to the year 2011. Canada applied for a waiver until 2013 to bring the preferential access into compliance with WTO rules. This WTO waiver expired in December 2013.
The new Trade and Development Agreement will hopefully take into account the varying levels of development and the vulnerabilities of Small Island developing states, and see more dynamic trade, investment and relations among the parties.
JUNE 2014: THE CRUCIAL MOMENT
The end of June could signal the re-invigoration and increased discussion, interaction and trade between Canada and the CARICOM block, or it could be the start of the very opposite. June 30, 2014 is the agreed date to end CARICOM-Canada trade negotiations – with or without an agreement. This means that although the negotiators have concluded six rounds of talks, nothing is agreed upon until everything is agreed on!
The Trinidad skyline. (Photo by Janique-ka John)
While the negotiations focus primarily on trade, the agreement is also a mechanism for reinforcing the relationship between the parties, which goes beyond trade and the text of the agreement. Canada-CARICOM relations go as far back as the 18th Century with the exchange of fish, lumber and the like for West Indian Rum, molasses and spices. Currently, Canada enjoys a close relationship with the Caribbean, particularly the Commonwealth countries therein, who have been supporters of Canadian positions in hemispheric and global issues. Canada is also home to a large portion of the Caribbean's diaspora, and Canadian investment in the region is significant and most notable in the financial sector.
Undoubtedly, through these negotiations Canada would want to strengthen its commercial relationship with the region and hopefully contribute to its development. CARICOM is seeking to increase Canadian investment in the region, improve market access for Caribbean goods and services in the Canadian market, as well as settle on comprehensive development cooperation.
From the onset though, the negotiations have been less than smooth, with both side unable to agree on the scope. Is this proposed agreement about international development or is it solely about trade? For countries in the Caribbean, and for north-south agreements in general, it has to be about both. For developing countries, trade agreements are useless without a framework to support capacity building and develop. At issue is also the extent to which Canada would want to espouse development commitments within an agreement. Doing so may set precedent for future negotiations, and this is a position that Canada may be trying to avoid.
The next issue is undoubtedly the architecture of the agreement, particularly as it relates to services and investment. Both sides have been known to use different approaches to scheduling liberalizations and reconciling this would not be an easy task. Also, sectoral carve-outs and whether or not there should be side-agreements and the chapters to be included would have been a hurdle. For example, should culture be a side agreement, or be part of the agreement at all?
Moreover, as with all trade negotiations the extent of liberalization is a sticking point, but perhaps the Economic Partnership Agreement (EPA) between CARIFORUM and the EU that was concluded before prejudices these negotiations. Under the EPA, if CARICOM countries negotiate a more favourable agreement with another major trading economy (and Canada qualifies as such under the EPA), they have to give that level of preference to the EU as well. CARICOM negotiators would therefore have to be mindful of the preferences that it offers to Canada, and the EPA in many cases would be the ceiling of their commitment to Canada. On the other hand too, the EPA is a comprehensive trade agreement, which contains specific commitments to development assistance and funding that is predictable and identifiable. If Canada expects the same (and even higher) levels of commitment from CARICOM as compared to the EPA, then CARICOM would naturally expect the same and even higher levels of commitments from Canada, particularly as it relates to development cooperation.
As if all these challenges were insufficient, negotiators now have to contend with a deadline. As the saying goes "hurry birds do not build good nests". On both sides, governments should be weary of trying to hurriedly come to a decision in order to meet a deadline. An agreement had not been reached as yet at the start of June. If one is reached by the end of the month it is always going to be called into question, particularly on the side of CARICOM who paradoxically has a lot to gain from the potential of what a sound agreement can do; and also the least to gain from a hurried ill-conceived compromise.
For this to be a truly successful negotiation that all parties can benefit from there needs to be flexibility and compromise on both sides. The longstanding issues as it relates to rum should be addressed in addition to the restrictive rules of origin from CARICOM products. The agreement should take into consideration the characteristics that make small-island developing states vulnerable, in that light facilitating asymmetrical liberalization and phased tariff cuts. It is important that it goes beyond market access to address supply side constraints and incapacities, assist in the transfer of technologies and knowledge. A sound agreement can undoubtedly bring new opportunities to both Caribbean and Canadian businesses.
Therefore, the region eagerly awaits what will happen post-June, particularly given the tone of negotiations currently.
CARICOM: Antigua and Barbuda; The Bahamas; Barbados; Belize; Dominica; Grenada; Guyana; Haiti; Jamaica; Montserrat; St. Lucia; St. Kitts and Nevis; St. Vincent and the Grenadines; Suriname; Trinidad and Tobago.
CARIFORUM: CARICOM and Dominican Republic.
Aurelia Bruce is a Research Officer at the TTCSI and has had experience conducting trade-related research in both the Caribbean and Canada. Bruce is passionate about regional integration and development and while her research centers around trade in services generally, some of her interests include the creative industries, ICT and ITeS, financial services, and trade law and policy. She graduated from the University of the West Indies (UWI), Cave Hill Campus, with a Master's degree in International Trade Policy (Distinction) and a BSc. in Political Science with Law (Hons).
Featured Photo by Aurelia Bruce.
Aurelia Bruce
MMiddle East
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redpajama_set_name: RedPajamaCommonCrawl


By Moses Michira -Standard Digital , Nairobi - Somalia Prime Minister Omar Abdirashid Sharmarke was yesterday detained for several hours in Nairobi for breaching protocol.
The embarrassing incident that could test diplomatic relations between Kenya and war-torn Somalia happened mid-morning yesterday at the Jomo Kenyatta International Airport.
Sharmarke could not be cleared in Nairobi because his plane had not been inspected at Wajir Airport, as is the requirement for all flights to and from Somalia. "This is a real breach of security," an official at JKIA told the Standard on Sunday. By 2 pm, security and foreign affairs officers had not agreed on how to handle the Somali leader. Security officers had demanded that Mr Sharmarke be sent back to Mogadishu, notwithstanding that the plane he had arrived in had left immediately after the entourage disembarked. But the department of protocol at the Ministry of Foreign Affairs had already provided the visiting head of government with a fleet of cars — which would remain parked at the VIP arrivals gate for several hours. The chartered plane carrying the Somali top official and his delegation touched down at 10.30 am, and left for Hargeisa in Somaliland. Officials at the JKIA disclosed that the jet was chartered from Freedom Airline Express. Mr Sharmarke was in a 20-member delegation of top officials of his government, including three ministers. The delegation was to early today make a connecting flight to an unnamed Middle East country, after a day-long stop-over in Nairobi. Kenya has enjoyed rather cordial relations with unstable Somalia, and is leading the fight against Al-Shabaab militia group that has ravaged the country. Some Somalis view the pursuit of the militia inside their country as an unwarranted invasion, a possible link to the recent mass-casualty attack staged in a base in El Adde. Somalia has dragged Kenya to an international court where it is demanding that international borders be re-drawn to grant it most of the Kenyan coastline. A decision is yet to be made on the same.


Injectable, in situ-gelling nanostructured hydrogels have been prepared from hydrazide and aldehyde-functionalized polymer precursors based on a copolymer of oligo(ethylene glycol) methacrylate (OEGMA) and an oligo(lactic acid) macromonomer (OLA) with varying lactic acid chain lengths. The resulting hydrogels contain a mix of chemical (hydrazone bond formation between hydrazide and aldehyde groups) and physical (hydrophobic interactions between OLA chains) cross-links which form competitively as a function of the OLA chain length and density. An increase in the OLA chain length and density results in the formation of more physical cross-links and fewer chemical cross-links. Tuning the relative prevalence of physical and chemical cross-link formation facilitated largely independent tuning of gel mechanics relative to gel swelling and degradation. Small-angle neutron scattering of these OLA-containing hydrogels reveals a microstructure consisting of associative hydrophobic domains, based on an increased scattering intensity and decreased blob size relative to that observed for POEGMA hydrogels prepared without the OLA co-monomer. The presence of hydrophobic OLA domains increases the uptake and slows the release of bovine serum albumin, a protein well-known to associate with hydrophobic domains. Coupled with the observed cytocompatibility of the reactive precursor polymers used to prepare the hydrogels, we anticipate significant potential applications of these hydrogels for the prolonged release of hydrophobic cargoes.


Cars Movie Gift Ideas for kids who love Lightning McQueen!
Confession time. I really feel I should get this weight off my chest before I show you the wonderful gifts you can get.
I used to be the person who pitied the parents with kids dressed up like a movie character.
I thought it showed a lack of imagination. If kids had dress up things EXACTLY like the movie then they wouldn't have space for real imagination.
My son uses "the yellow school bus" Cars 3 movie toy from a chocolate Kinder surprise egg very happily since I refused to buy him anything larger.
Honestly though, it doesn't matter to him that it is not the same size as his other Cars movie vehicles. It is the correct vehicle and that's all that matters to him. His imagination makes it the right size… and fills in the storyline as the cars race around doing whatever it is that cars do when they have feelings and faces and talk and all.
So now I encourage any imaginative play. And if that means special vehicles to help it happen, then I don't mind!
So let's start with the vehicles since that is the most important.
First up it must be the hero of course.
And a race track. So Lightning McQueen can win again and again… or not.
If you have read any of my other posts here you will know that I am a big Lego and Duplo fan. So this Duplo Piston cup race looks perfect to me!
My now nearly 6-year-old son has this Lego set, and after a year it is still a favorite. He has enjoyed building it and rebuilding it, and making sure that Cruz is nicely tuned up and ready for action… It really is a good bridge between Duplo and "big kid" lego.
Now this set is for those kids that really want to be mechanics. There are all the tools to fix Lightning up as good as new if he crashes while racing!
My son loves "the school bus" from the third movie. He would adore this.
The faster the cars go, the better they are (so I am told – what do I know driving the ultimate family car that smells and is complete with unknown sticky substances for you to accidentally put your hand into when trying to find your kid's seatbelt).
This would definitely be a winner on the speed front!
And of course a garage will be very necessary.
With the cars to go in both, of course.
Mack is probably my favorite character. Although to be fair, it is very hard to choose from a great line-up. This version of Mack transforms into a raceway. Even I want to play with it.
My nephew prefers Jackson Storm. I think it may be the color. Either way he would love to open this on Christmas day.
There are some really inventive games out there, using the Disney Cars characters. I was really impressed. I hope you like this selection. Any of these would make my kids so happy!
I found a selection of books that I think any fan would enjoy.
Of course, this list has to have a book that makes sounds.
No little Cars fan can be happy without suitable clothing to wear with their awesome vehicles.
For those fathers who want to support their kids, or who also really like the movies, there are these that you could use to make for some cute father and son gifts.
Now to move on to the décor. After all it is very important to have everything looking the part. Lightning McQueen has his number and colors all over his things so you need to do the same for little Lightening McQueens, or Jackson Storms, or Cruz Ramirez.
But if they saw this cake tin, I suspect that it would win for the next 2 birthdays at least.
And there are candles to match! It's so cute!
So now I feel I can leave you having given you some awesome options for any kids addicted to the Cars movies.


Architect Axel Stein has designed this luxury boutique hotel in Germany as a contemporary version of the classic Alpine chalet with a mix of warm, welcoming ambiance and cool modern style. We like the uniquely crafted architectural angles, the rustic natural materials and colorful accents, and the stunning Alpine views.
Set among the Alpine valleys in southern Bavaria, this luxury boutique hotel in Germany offers the exclusivity of a private hideaway coupled with the service, amenities and hospitality of a four star hotel and wellness spa.
Inspired by the idea of opposites – Berg (mountains) und Tal (valleys) – the concept was developed by the current owners in partnership with architect Axel Stein to create a unique combination of traditional 'Allgäuer' style paired with modern design. Consisting of eight luxury lofts, the hotel was remodeled from an earlier hotel dating back to the 1960s and features the classic roof lines and traditional structural shapes of the region. Remodeled with locally sourced timber and stone along with sleek modern materials and finishes, the effect is a unique fusion of rustic and contemporary.
The interiors were redesigned by the owners with a mix of designer furnishings and hand selected decor with natural fabrics and bolts of lively color creating a stylish but relaxing atmosphere. Each loft comes complete with a fully equipped kitchen and double en suite bedroom, and all feature panoramic views of the Allgäu valley and mountains. Lofts one, two and three share private gardens and each have access to the wellness spa and hotel restaurant.
Breakfast delivered daily in your own breakfast box or, if preferred, a breakfast buffet is served in the Schlossanger Alp. Each loft fridge is stocked with milk, butter, eggs and other small ingredients for cooking as well as a Nespresso coffee maker with a choice of coffees and Ronnefeldt tea. There is also an "honesty bar" wine cabinet stocked with a favorite selection from the owners. Use of the spa which includes pool, whirlpool spa, steam, sauna and optional massage, physiotherapy and beauty treatments, bath robes and spa bag with everything you need to relax. Housekeeping provided every fourth day. Chauffeur service from the lofts to the Schlossanger Alp or Falkenstein provided according to schedule. Wifi throughout, home entertainment system/TV with iPod audio dock. Each loft comes with one private garaged parking spot and storage box for skis, bikes and other outdoor gear.
Nestled at the foot of the Bavarian Alps in south western Germany, the Allgau region stretches from the lower alpine valleys to the upper alpine slopes. A picturesque landscape of lakes and mountains, the area is popular as a spa and winter sport destination as well as for King Ludwig's famous Schloss Neuschwanstein castle just a short drive away. The slopes of Breitenberg and Skizentrum Pfronten are within a short drive and the charming town of Pfronten offers great small town life with good restaurants and shops.


text:
Belle Epoque brooch/pendant
Le Saché, Georges
A beautiful Belle Epoque platinum ajour, diamond and pearl pendant/brooch by Le Saché.
Makers mark: LS
Size: 5x5.5 cm
LE SACHE, GEORGES Le Saché, Georges (b. 1849 - ?) Georges le Saché was an important jewellery designer and manufacturer who came from a family of artists and engravers. His father, Emilé Le Saché, was a draughtsman and line engraver. His mother had a modest business selling jewellery at the Palais Royale. 1866 - Le Saché spent his youth in painter's studios, desperate to become an artist. His parents, seeking a more stable career for their son, sent him to Germany to join Friedman: an important German jeweller. Following this initial experience in the jewellery business, Le Saché went to London to apply his artistic talents in various areas of the decorative arts. 1870 - Outbreak of the Franco-Prussian war sees Le Saché returning home and joining the 1st Battalion of the Mobile Seine Regiment. He fought bravely between September 1870 and March 1871. Following the war Le Saché returned to London. 1872 - Le Saché joins renowned Parisian jewellers Falize as a designer and collaborator. He spent five years with Lucien Falize learning the jeweller's craft and becoming a skilled manufacturer 1877 - Le Saché marries the daughter of Baucheron, who was a partner in the firm Baucheron & Guilian, one of the most important manufacturing jewellers of the period. Le Saché works for the family firm and eventually takes over the business. Le Saché became one of the most sought-after manufacturers in Paris, supplying all the best jewellery houses with their prime exhibits at the various International Exhibitions. He worked in both bijouterie and joaillerie, producing a rather formal version of Art Nouveau. Le Saché ran his own workshop for 30 years and always kept his works anonymous. 1901 Georges Le Saché was awarded a silver gilt plaque by the Chambre Syndicale de la Bijouterie to recognise his distinction as a jeweller and designer. This was the highest accolade which was richly deserved. 1920 - Georges Le Saché's makers mark (LS within a horizontal lozenge) was deleted on 6th January 1920
Cachet smokey quartz


text:
Robert Kirkman Mission Statement: Fix Comic Industry
Video Editorial: Robert Kirkman I came back from San Diego supercharged with excitement for comics and so I made this video. It's a little ramble filled in places and I want to thank the fine folks at CBR for cleaning it up as much as they did. Just for clarity's sake I'd like to simplify...
brian michael bendis robert kirkman
Forum: The Lounge
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Matt Fraction twittered this: Robert kirkman is now a partner at image. http://www.nytimes.com/2008/07/22/books/22kirk.html?_r=2&oref=login&oref=slogin Kirkman was marvel exclusive apart from Walking dead and Invincible....then he started that wolfman series....anyway the contract finished...
Ninja4peace
image robert kirkman
Forum: General Comics
McFarlane & Kirkman!
They really are working on something after all. From Lying in the Gutters: KIRKMANIA "I understand a Todd McFarlane/Robert Kirkman project will be announced at the Todd McFarlane panel at SDCC."
robert kirkman todd mcfarlane
The Walking Dead (Image) [SPOILERS]
from some of the comments that i've heard, a few people do pick up the comic The Walking Dead by Robert Kirkman. so far up to issue 21 the story begins with police officer Rick Grimes, who wakes from a long coma to find his hospital bereft of staff or patients, and littered with the living...
robert kirkman walking dead
I've always heard about this comic, but have never really shown much interest in it, until today when I stumbled onto the first issue online. A really light hearted read, with amazing artwork, very reminiscent of Stuart Immonen, one of my favourites. So, I'm strongly considering picking up the...
Fuzzy Birds


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