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Partial Identification under Imperfect Measurement: A Distributionally Robust Approach
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\maketitle
\begin{abstract}
Imperfect measurements allow different underlying distributions to generate the same observations.
Restrictions linking unobserved components can make it difficult to characterize all compatible distributions and compute the resulting range of parameter values.
We ask how to construct computable parameter bounds and tighten them without additional data or stronger assumptions.
We develop a general framework for partial identification under imperfect measurement based on distributionally robust optimization.
We use measurement restrictions to construct a neighborhood around a benchmark distribution, ensuring that every compatible distribution lies within it.
Optimizing over this neighborhood gives parameter bounds through a common procedure that can be adapted to different measurement problems.
The choice of benchmark affects how informative these bounds are.
We establish conditions under which changing it can tighten the bounds while preserving every parameter value consistent with the same data and assumptions.
Applications to networks, missing outcomes, and regression illustrate how the framework tightens bounds and can recover the full range of compatible values.
\end{abstract}
\vspace{1.5\baselineskip}
\noindent\textit{Keywords:} Partial identification; imperfect measurement; distributionally robust optimization; Wasserstein distance; benchmark selection.
\clearpage
\begin{bibunit}
\def.#1{.main}
\def.#1{.main}
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\section{Introduction}
\label{sec:introduction}
Economic parameters often depend on information that the data reveal only partially.
Outcomes may be recorded only for selected individuals, or a reported network may conceal the strength of interactions.
Different explanations for this missing information can imply different population means or structural effects, even when the distribution of the observations is known.
The sharp identified set collects all values consistent with the observations and maintained assumptions \cite{Manski1989}.
Computing it can be difficult when structural restrictions link unobserved components.
We ask how to construct computable outer bounds from the measurement model and systematically tighten them by selecting a benchmark distribution, while preserving every parameter value allowed by the same observations and identifying assumptions.
Our primary contribution is a general framework for partial identification under imperfect measurement based on distributionally robust optimization (DRO).
Starting from a benchmark latent distribution, such as an imputation, we use measurement restrictions to bound its distance from every compatible distribution.
The benchmark serves as the center of a neighborhood containing all distributions consistent with the data and assumptions.
Intuitively, the Wasserstein distance measures the smallest average change in latent values needed to transform one distribution into another.
The neighborhood's radius sets the limit on this change.
This neighborhood gives computable parameter bounds.
Choosing its center carefully can tighten both the neighborhood and the resulting bounds while retaining every compatible distribution.
Researchers specify the measurement model, the parameter of interest, and methods for solving the required optimization problems.
The framework then follows the same steps to construct bounds, allowing the procedure to be automated and reused across applications.
Our second contribution establishes when changing the benchmark yields bounds that are at least as tight as the original ones and still contain every parameter value consistent with the data and assumptions.
Theorem~\ref{main:recentering} and Corollary~\ref{main:triangle-certificate} provide both a simple rule for ensuring this guarantee and a more general test that allows additional benchmark choices.
We use these results to formulate optimization programs for benchmark selection.
We study scalar parameters defined by population moments, taking the observed law as known.
The main results use a compact latent space, continuous moments, and a closed, convex class of retained laws.
The framework allows nonsmooth and noninvertible measurement rules.
The approach can save computation when compatibility restrictions are easier to use in calibrating the neighborhood than to impose directly in moment optimization.
Our network application shows how to refine bounds on a moment-defined network-exposure slope when restrictions on which hidden links can coexist make the sharp set difficult to compute.
Known limits on unreported link weights give valid initial bounds, and benchmark selection tightens them without additional data or stronger identifying assumptions.\footnote{
Network-measurement research studies sampled networks \cite{ChandrasekharLewis2016}, censored nominations \cite{Griffith2022}, unobserved links \cite{LewbelQuTang2023}, and network identification from panel outcomes \cite{dePaulaRasulSouza2025}.}
The missing-outcome and regression examples show that the same procedure can also recover the sharp identified set.
\subsection{Related work}
\label{sec:related-work}
Our use of distributional neighborhoods builds on robust bounds for counterfactuals and treatment effects under uncertainty about latent distributions, marginals, or covariate noise \cite{ChristensenConnault2023,GuRussell2024,FanParkXu2025,GuoEtAl2022}.
The question here is how to choose such a neighborhood for a fixed measurement model.
We calibrate its radius to cover every compatible law, making the benchmark a computational choice that can change without altering the identifying assumptions.
This containment requirement connects our construction to random-set, latent-variable, and transport methods that characterize compatibility with incomplete observations \cite{BeresteanuMolchanovMolinari2011,BeresteanuMolchanovMolinari2012, EkelandGalichonHenry2010,Schennach2014}.
Imposing the full compatible class can remain computationally demanding.
We use measurement restrictions to ensure that a simpler class used for moment optimization still contains every compatible law.
The requirement to retain every compatible law also distinguishes our objective from transport-based moment fitting, which seeks a least-cost adjustment satisfying the moments \cite{SchennachStarck2026}.
Turning these compatibility conditions into bounds is the focus of transport methods and automated polynomial optimization, which provide sharp identified sets or certified approximations \cite{FanParkPassShi2025,Voronin2025,FranguridiLiu2026,DuarteEtAl2024}.
Existing approximation and refinement guarantees do not by themselves justify changing a distributional benchmark.
We certify such changes by requiring the new enclosing class to lie inside the original one.
The resulting guarantee of no wider bounds applies to every target moment covered by the framework.
Selecting that benchmark introduces a geometric problem studied by enclosing-ball methods, which choose centers by minimizing a covering radius \cite{MordukhovichNamVillalobos2013,WangEtAl2025}.
A smaller radius alone does not ensure tighter parameter bounds.
Our nesting conditions use the retained restrictions to determine when a change in center also refines the envelope.
They can therefore certify refinements that a comparison of the unrestricted balls would miss.
Our framework links these ideas through benchmark choice: measurement restrictions determine what the envelope must contain, and nesting conditions determine which changes can refine the moment bounds.
The benchmark therefore governs the computational approximation, while the observed law, identifying assumptions, and sharp identified set remain fixed.
\paragraph{Organization.}
Section~\ref{sec:model} presents the model, Section~\ref{sec:main-results} develops and applies the framework, and Section~\ref{sec:conclusion} concludes.
Appendices A--C provide proofs and computational details; D--F develop the missing-outcome, network, and regression applications.
\section{Model: Learning from imperfect measurements}
\label{sec:model}
\subsection{The parameter and the measurement rule}
We study what can be learned about a population parameter when the variables defining it are only partly observed.
Let $X^\star$ denote the complete, latent data and let $P_{X^\star}$ denote their true probability law.
A scalar parameter $\theta_0\in\Theta\subseteq\mathbb R$ is characterized by
\begin{equation}
\mathbb E_{P_{X^\star}}[g_\theta(X^\star)]=0
\quad\text{if and only if}\quad \theta=\theta_0.
\label{eq:true-moment}
\end{equation}
The known function $g_\theta$ specifies a population relationship.
For a mean, for example, $g_\theta(X^\star)=\theta-Y$.
A regression slope is defined by an analogous moment involving both the outcome and the regressor.
We observe $X$, generated by a known measurable measurement rule
\begin{equation}
X=f(X^\star).
\label{eq:measurement-map}
\end{equation}
The map $f$ may combine a true value with measurement error, discard a missing outcome, or replace an interaction weight by a binary report.
We write $\mathcal X^\star$ and $\mathcal X$ for the latent and observed spaces, and $P_X$ for the observed probability law.
This paper takes $P_X$ as known and studies \emph{population identification}.
Estimation from a finite sample is a subsequent question.
With random measurement error, we include signal and noise in $X^\star=(\text{signal},\text{noise})$, so $f$ remains deterministic given the complete state, as illustrated in Appendix~\ref{app:measurement-error}.
Appendix~\ref{app:stochastic} gives the alternative formulation with stochastic measurement and specifies the required error restrictions.
A noninvertible $f$ can leave several latent explanations for the same observations.
Whether those explanations imply different parameter values depends on the target and the maintained assumptions.
\subsection{Compatible laws and the identified set}
Let $\mathcal P(\mathcal X^\star)$ denote the probability laws on the latent space.
Collect the maintained support and other restrictions in a class $\mathcal C\subseteq\mathcal P(\mathcal X^\star)$.
These may include bounds on network weights, restrictions on measurement error, or conditions ensuring that a regression slope is well-defined.
We assume the true law belongs to $\mathcal C$ and retain any conditions needed for the moment to define the target uniquely.
All moments below are assumed to exist.
\begin{definition}[Compatible latent laws]
\label{def:exact-law-set}
The laws consistent with the observed population are
\begin{equation}
\mathcal Q_f(P_X)=\{Q\in\mathcal C:f_\#Q=P_X\}.
\label{eq:compatible-laws}
\end{equation}
Here $f_\#Q$ means the distribution obtained by applying $f$ to data with law $Q$.\footnote{Formally, $f_\#Q$ is the pushforward of $Q$ under $f$: $(f_\#Q)(A)=Q(f^{-1}(A))$ for every measurable $A\subseteq\mathcal X$.}
Thus a compatible law reproduces the \emph{entire} observed distribution, including the joint distribution of any variables measured without error.
\end{definition}
We assume $\mathcal Q_f(P_X)$ is nonempty when describing its identified parameters.
\begin{definition}[Exact identified set]
\label{def:exact-id}
The exact population identified set is
\begin{equation}
\Theta_{\mathrm{ID}}(P_X)
=\{\theta\in\Theta:\exists Q\in\mathcal Q_f(P_X)
\text{ such that }\mathbb E_Q[g_\theta(X^\star)]=0\}.
\label{eq:identified-set}
\end{equation}
\end{definition}
Each value in this set has a compatible latent explanation.
Values outside it cannot satisfy the stated model.
This is the meaning of \emph{sharp identification}.
A singleton gives point identification; a larger set gives partial identification.
The set need not be an interval in general.
Figure~\ref{fig:model-geometry} summarizes the two steps: identify admissible laws, then retain the parameter values they can support.
Additional structural restrictions must also be imposed when claiming sharpness for a stronger model.
The network example illustrates this distinction.
\begin{figure}[htbp]
\centering
\makebox[\linewidth][c]{\scalebox{1.1}{
\begin{tikzpicture}[
>=Latex,
line cap=round,
line join=round,
font=\Large,
every node/.style={inner sep=1.5pt}
]
\definecolor{modelteal}{HTML}{157A78}
\draw[black!45,line width=1pt] (1.48,0.68) ellipse [x radius=1.48, y radius=1.10];
\node[text=black!75] at (0.10,1.92) {\large$\Theta$};
\begin{scope}[shift={(1.42,0.36)}, rotate=28]
\fill[inkblue!21] (0,0) ellipse [x radius=0.84, y radius=0.48];
\draw[inkblue,dashed, line width=0.7pt] (0,0) ellipse [x radius=0.84, y radius=0.48];
\end{scope}
\node[text=inkblue] at (0.88,0.97) {\normalsize$\Theta_{\mathrm{ID}}$};
\coordinate (theta) at (1.72,0.45);
\fill[inkblue] (theta) circle (1.7pt);
\node[text=inkblue,below left=1pt and 0.3pt of theta] {\small{$\theta_0$}};
\filldraw[fill=black!6,draw=black!45,line width=0.9pt] (4.55,-0.45) rectangle (9.35,1.95);
\node[text=black!75] at (8.82,2.22) {\normalsize$\mathcal{P}(\mathcal{X}^{\star})$};
\begin{scope}[shift={(6.80,0.65)},scale=1.30]
\path[fill=inkblue!21,draw=inkblue,line width=1.05pt]
(-.80,-.10)
.. controls (-.95,.22) and (-.67,.69) .. (-.33,.72)
.. controls (-.05,.75) and (-.11,.44) .. (.16,.50)
.. controls (.46,.66) and (.68,.66) .. (.75,.32)
.. controls (.77,.14) and (.63,.06) .. (.69,-.21)
.. controls (.87,-.57) and (.31,-.78) .. (.03,-.61)
.. controls (-.27,-.41) and (-.52,-.83) .. (-.66,-.47)
.. controls (-.74,-.28) and (-.70,-.24) .. (-.80,-.10)
-- cycle;
\end{scope}
\node[text=inkblue] at (6.72,0.23) {\normalsize$\mathcal Q_f(P_X)$};
\coordinate (pxs) at (7.05,0.82);
\fill[inkblue] (pxs) circle (1.7pt);
\node[text=inkblue,above=1pt of pxs] {\normalsize$P_{X^\star}$};
\filldraw[fill=black!6,draw=black!45,line width=0.9pt] (10.35,-0.45) rectangle (14.65,1.95);
\node[text=black!75] at (14.42,2.22) {\normalsize$\mathcal{P}(\mathcal{X})$};
\coordinate (px) at (11.45,0.70);
\fill[modelteal] (px) circle (1.7pt);
\node[text=modelteal,above right=2pt and 4pt of px] {\normalsize$P_X$};
\draw[black!65,line width=0.9pt]
(theta) .. controls (3.10,1.55) and (5.10,0.55) .. (6.88,0.82);
\node[text=black!75] at (3.55,2.22) {\small{$\mathbb E_Q[g_\theta(X^\star)]=0$}};
\draw[modelteal,->, line width=0.9pt]
(7.18,0.86) .. controls (8.75,1.18) and (9.95,1.10) .. (11.28,0.75);
\node[text=modelteal] at (9.80,1.3) {\small{$f_\#$}};
\draw[inkblue,dotted, ->, line width=0.9pt]
(11.34,0.48) .. controls (9.80,0.02) and (8.55,0.04) .. (6.85,0.54);
\end{tikzpicture}
}}
\caption{The solid $f_\#$ arrow maps the true latent law $P_{X^\star}$ to $P_X$.
The dotted arrow denotes the compatible-law correspondence $P_X\mapsto\mathcal Q_f(P_X)$.
The left curve denotes the moment relation $\mathbb E_Q[g_\theta(X^\star)]=0$.
The shaded, dashed region is the identified set $\Theta_{\mathrm{ID}}(P_X)$.}
\label{fig:model-geometry}
\end{figure}
\subsection{Two examples}
\begin{example}[Missing outcomes]
\label{ex:manski-missingness}
Let $D=1$ indicate participation in the workforce and let $Y\in\{0,1\}$ indicate income above a threshold.
Income is observed only for workers.
The target is the population mean $\theta_0=\mathbb E[Y]$, with $\Theta=[0,1]$ and
\[
X^\star=(D,Y),\qquad X=f(D,Y)=(D,DY),\qquad
g_\theta(D,Y)=\theta-Y.
\]
Both latent states $(0,0)$ and $(0,1)$ produce the observed state $(0,0)$.
Write $p_{ab}=P_X\{(a,b)\}$.
The observed law fixes $p_{10}$ and $p_{11}$, but leaves the success probability among nonworkers unrestricted.
Let $Q_t$ be a possible distribution of $(D,Y)$ that preserves the observed worker outcomes and group proportions, while allocating a fraction $t\in[0,1]$ of the nonworker group to $Y=1$.
Then
\begin{equation}
\mathbb E_{Q_t}Y=p_{11}+p_{00}t,
\qquad \Theta_{\mathrm{ID}}(P_X)=[p_{11},p_{11}+p_{00}].
\label{eq:missing-sharp}
\end{equation}
For example, $p_{00}=0.2$ and $p_{11}=0.4$ give the interval $[0.4,0.6]$.
This familiar missing-outcome bound \cite{Manski1990,HorowitzManski2000} is directly computable because a single unknown probability $t$ describes all compatible laws.
It provides a transparent benchmark for evaluating the later robust construction.
The full law and sharpness argument are in Appendix~\ref{app:missingness}.
\end{example}
\Needspace{4\baselineskip}
\begin{example}[Network-exposure slope with a thresholded network]
\label{ex:lim-threshold}
There are $m\ge2$ treated core individuals, indexed $1,\ldots,m$, and two additional individuals: the treated contact $h$ and the untreated contact $s$.
Thus $n=m+2$.
Let $Y\in\mathbb R^n$ collect their outcomes and $T\in\{0,1\}^n$ their treatment indicators, so $T_1=\cdots=T_m=T_h=1$ and $T_s=0$.
The latent network $G^\star\in[0,1]^{n\times n}$ records peer weights: $G^\star_{ij}$ is the weight individual $i$ places on individual $j$'s treatment.
A known set $E$ specifies eligible pairs among the core individuals, oriented from the smaller to the larger index.
Each pair has a known weight $w_{ij}>0$ and capacity $\eta w_{ij}$, where $\eta>0$ is a known scale.
Individual $i$ has an unobserved role $z_i\in\{0,1\}$, distinct from treatment status.
An eligible link is active only between different roles.
Define
\[
\begin{aligned}
G^\star_{ij}
&=\eta w_{ij}\mathbf1\{z_i\ne z_j\}
&&((i,j)\in E),\\
G^\star_{is}
&=1-\sum_{j:(i,j)\in E}G^\star_{ij}
&&(i=1,\ldots,m),\\
G^\star_{h1}&=G^\star_{s1}=1,
\end{aligned}
\]
with all other entries zero.
Each core individual allocates a unit of weight between eligible treated peers and the untreated contact $s$.
An active peer link therefore reduces the weight on $s$.
The last line specifies the rows of the two additional individuals.
Outcomes and treatments are observed without error.
The network is reported only through a known threshold $0<\xi<1$:
\[
G=\mathbf1\{G^\star\ge\xi\}\quad\text{entrywise}.
\]
Choose $\eta$ so that $\eta\max_{i\le m}\sum_{j:(i,j)\in E}w_{ij}<\min\{\xi,1-\xi\}$.
Then every eligible peer weight is below $\xi$, every core weight on $s$ exceeds $\xi$, and each row sums to one.
For fixed contact labels, every hidden-role assignment therefore gives the same reported network.
A motivating outcome model is
\[
Y=\beta_0T+\theta_0G^\star T+\varepsilon,
\qquad \mathbb E[\varepsilon\mid G^\star,T]=0.
\]
We take $\beta_0$ as known to isolate network uncertainty.
We maintain the scalar moment below and graph restrictions, but not the conditional mean restriction.\footnote{See Appendix~\ref{app:network}.}
In the general notation, the latent state and measurement map are
\[
X^\star=(Y,T,G^\star),\qquad
X=f(X^\star)=(Y,T,\mathbf1\{G^\star\ge\xi\})=(Y,T,G),
\]
and the target $\theta_0\in\Theta=\mathbb R$ is defined by the moment
\begin{equation}
g_\theta(Y,T,G^\star)
=T^\top(Y-\beta_0T)-\theta T^\top G^\star T.
\label{eq:network-moment}
\end{equation}
The maintained class $\mathcal C$ imposes the treatment pattern and graph construction above.
Hidden roles may depend arbitrarily on the observed record.
We take the joint law $P_X$ of $(Y,T,G)$ as known; compatible laws $Q\in\mathcal Q_f(P_X)$ additionally reproduce this entire law.
Consequently,
\[
\mathbb E_Q[g_\theta(X^\star)]
=\mathbb E_{P_X}[T^\top(Y-\beta_0T)]
-\theta\mathbb E_Q[T^\top G^\star T].
\]
The first expectation is known, although outcomes and contact labels may vary across observations.
Exposure $T^\top G^\star T$ is the total network weight between treated units.
The fixed link $G^\star_{h1}=1$ joins two treated individuals, so exposure is at least one.
Its positive expectation ensures that the moment defines a unique slope under each compatible law.
The shared roles restrict which hidden links can coexist.
Maximizing their total exposure can require a difficult global search, although the reported network is unchanged.
The later calculations replace this search with a coverage certificate and constrained moment optimization; explicit endpoint formulas validate the results.
Appendix~\ref{app:network} gives the compatible laws and the numerical specification.
\end{example}
\subsection{Informative bounds from imperfect measurements}
\label{sec:research-question}
Ideally, we would compute the sharp identified set directly.
It consists of all parameter values consistent with the observed law and the maintained measurement model.
This set describes exactly the uncertainty left by imperfect observation.
Our target is the parameter, so recovering this set does not require reconstructing every feature of the latent law.
The difficulty is that compatible laws may be hard to describe or optimize over.
A complicated measurement rule can leave many latent explanations that must reproduce the full observed distribution and satisfy the maintained restrictions.
Searching over these distributions can make exact parameter bounds computationally impractical, even when the target is a single mean or slope.
We propose a relaxation based on distributionally robust optimization (DRO) and transport geometry.
We allow enough variation around a benchmark latent law to include every compatible law, while retaining restrictions that are convenient to enforce.
Optimizing the moment over this structured class gives a \emph{DRO outer identified set} containing the sharp set.
The aim is to make computation more tractable while controlling the extra parameter uncertainty introduced by the relaxation.
Geometrically, this larger class of latent laws forms an envelope containing every compatible law.
A better choice of center can reduce the extra width it creates in parameter space, even if the envelope still contains additional laws.
The question is how to construct and refine computable parameter bounds from the measurement model while preserving every value consistent with the data and maintained assumptions.
\section{Main results: Computing the DRO identified set}
\label{sec:main-results}
We answer the question in Section~\ref{sec:research-question} by showing how to compute the parameter bounds implied by the proposed envelope and how to reduce their excess width.
Theorem~\ref{main:feasibility} computes the parameter values supported by laws in the envelope.
Theorem~\ref{main:tube} gives a simpler bound by controlling how far the expected moment can move from its benchmark value.
It also provides an exact alternative calculation when only support and distance restrictions define the envelope.
Theorem~\ref{main:recentering} gives conditions under which changing the benchmark keeps the new envelope inside the original one while retaining every compatible law.
This can tighten the parameter bounds without excluding values consistent with the data and maintained assumptions.
The examples illustrate each result and show when recentering tightens the bounds.
Proofs of the three main theorems and their supporting results are in Appendix~\ref{app:main-proofs}.
\subsection{The ambiguity model and its assumptions}
Let $Q_0$ be a benchmark law on the same space as the candidate latent laws.
For example, imputing values for missing outcomes defines such a law.
When the observed and latent spaces differ, using $P_X$ requires an explicit embedding or projection, as in the regression application in Appendix~\ref{app:measurement-error}.
The Wasserstein distance measures the least average movement between two laws:
\begin{equation}
W_1(Q,Q_0)=\inf_{\pi\in\Pi(Q,Q_0)}\int d(x^\star,z^\star)
\,\pi(dx^\star,dz^\star).
\label{eq:wasserstein-distance}
\end{equation}
Here $\Pi(Q,Q_0)$ is the set of joint laws with marginals $Q$ and $Q_0$; each joint law describes how mass moves between latent states.
The distance $d$ specifies which changes are costly.
\Needspace{4\baselineskip}
\begin{assumption}[Compact space and continuous moments]
\label{ass:ambient-regularity}
The latent space $(\mathcal X^\star,d)$ is a compact metric space, and $g_\theta:\mathcal X^\star\to\mathbb R$ is continuous for each $\theta\in\Theta$.
\end{assumption}
\begin{assumption}[Valid convex relaxation]
\label{ass:retained-class}
The class $\mathcal H\subseteq\mathcal P(\mathcal X^\star)$ is weakly closed and convex, contains $\mathcal Q_f(P_X)$, and includes $Q_0$.
It does not depend on $\theta$.
\end{assumption}
The class $\mathcal H$ retains restrictions we can conveniently enforce, such as network support or known observable marginals.
It may be all laws on the chosen space.
Closedness keeps limits admissible; convexity allows mixtures of admissible laws.
Together with compactness, these conditions ensure that the minimum and maximum moments are attained and that every intermediate value is attainable.
They do not require $f$ to be continuous or the exact compatible class to be convex.
For computation, we combine the retained restrictions with a transport budget $\delta\ge0$ around the benchmark:
\begin{align}
\mathcal A_\delta(Q_0)
&=\{Q\in\mathcal H:W_1(Q,Q_0)\le\delta\},
\label{eq:ambiguity-class}\\
\Theta_{\mathrm{DRO}}(\delta;Q_0)
&=\{\theta\in\Theta:\exists Q\in\mathcal A_\delta(Q_0),
\ \mathbb E_Q[g_\theta(X^\star)]=0\}.
\label{eq:dro-set}
\end{align}
We call $\mathcal A_\delta(Q_0)$ the \emph{envelope}: it is the class of latent laws over which we optimize.
The set $\Theta_{\mathrm{DRO}}(\delta;Q_0)$ collects the parameter values these laws can support.
To calibrate the budget and assess individual candidates, define the covering radius and the feasibility cost, respectively:\footnote{We use the convention $\inf\varnothing=+\infty$.}
\begin{align}
r^\star(Q_0)&=\sup_{Q\in\mathcal Q_f(P_X)}W_1(Q,Q_0),
\label{eq:covering-radius}\\
R(\theta;Q_0)&=\inf_{\substack{Q\in\mathcal H\\\mathbb E_Q[g_\theta(X^\star)]=0}}
W_1(Q,Q_0).
\label{eq:feasibility-profile}
\end{align}
A budget $\delta\ge r^\star(Q_0)$ includes every compatible law, so the resulting parameter set contains the sharp identified set.
A computable upper bound on $r^\star$ suffices for this guarantee.
The cost $R(\theta;Q_0)$ measures the least movement from $Q_0$ to a law in $\mathcal H$ satisfying the candidate moment.
Theorem~\ref{main:feasibility} shows that comparing this cost with $\delta$ gives an alternative way to compute the parameter set.
The envelope intersects the benchmark ball with the retained class $\mathcal H$.
Covering all compatible laws preserves their zero-moment witnesses, while the additional laws can introduce additional parameter values.
Figure~\ref{fig:algorithm-recentering} shows this geometry together with benchmark refinement.
\subsection{Two ways to compute feasible parameters}
For a chosen benchmark and radius, the next two results turn the envelope into parameter bounds.
Theorem~\ref{main:feasibility} uses the smallest and largest expected moments allowed by the envelope to determine exactly which parameters belong to the DRO set.
Under an additional smoothness condition, Theorem~\ref{main:tube} gives a simpler outer bound using the benchmark moment and radius.
When support and distance are the only law restrictions, it also gives an exact optimization formula for the moment bounds.
These calculations supply the inputs to the parameter-set computation in Section~\ref{sec:parameter-computation}.
\Needspace{4\baselineskip}
\begin{theorem}[General computation of the DRO set]
\label{main:feasibility}
Under Assumptions~\ref{ass:ambient-regularity} and \ref{ass:retained-class}, the attainable values of $\mathbb E_Q[g_\theta(X^\star)]$ over $\mathcal A_\delta(Q_0)$ form a closed interval.
For every $\theta\in\Theta$, the following are equivalent:
\begin{equation}
\begin{aligned}
\textnormal{(i)}\quad&\theta\in\Theta_{\mathrm{DRO}}(\delta;Q_0);\\[2pt]
\textnormal{(ii)}\quad&
\min_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[g_\theta(X^\star)]
\le0\le
\max_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[g_\theta(X^\star)];\\[2pt]
\textnormal{(iii)}\quad&R(\theta;Q_0)\le\delta.
\end{aligned}
\label{main:feasibility-test}
\end{equation}
If $\delta\ge r^\star(Q_0)$, then $\Theta_{\mathrm{ID}}(P_X)\subseteq \Theta_{\mathrm{DRO}}(\delta;Q_0)$.
If also $\mathcal H=\mathcal Q_f(P_X)$, the two parameter sets are equal.
\end{theorem}
The computation requires two moment optimizations for each candidate $\theta$, or one minimum-cost calculation.
If zero lies between the two bounds, including equality, mixing their laws yields a law with zero moment.
If both bounds are strictly positive or strictly negative, that parameter is ruled out.
This is exact for the chosen DRO model.
With a valid covering radius, any excess over the sharp set comes from additional laws admitted by the envelope.
\Needspace{4\baselineskip}
\begin{assumption}[Moment smoothness]
\label{ass:moment-sensitivity}
For each fixed $\theta\in\Theta$, the moment function is Lipschitz continuous in the latent state, with a known finite constant $L(\theta)\ge0$:
\begin{equation}
|g_\theta(x^\star)-g_\theta(z^\star)|
\le L(\theta)d(x^\star,z^\star)
\quad\text{for all }x^\star,z^\star\in\mathcal X^\star.
\label{eq:moment-lipschitz}
\end{equation}
\end{assumption}
The smoothness condition bounds how far expected moments can move from the benchmark, giving a simpler outer bound without solving the moment optimizations in Theorem~\ref{main:feasibility}.
When only support and distance restrictions are retained, the next result also gives a dual formula for the exact moment bounds.
\Needspace{4\baselineskip}
\begin{theorem}[A simpler bound under smoothness]
\label{main:tube}
Under Assumptions~\ref{ass:ambient-regularity}--\ref{ass:moment-sensitivity}, every $Q\in\mathcal A_\delta(Q_0)$ satisfies $|\mathbb E_Q[g_\theta(X^\star)]-\mathbb E_{Q_0}[g_\theta(X^\star)]| \le\delta L(\theta)$.
Consequently,
\begin{equation}
\Theta_{\mathrm{DRO}}(\delta;Q_0)
\subseteq\mathcal T_\delta(Q_0)
:=\{\theta\in\Theta:|\mathbb E_{Q_0}[g_\theta(X^\star)]|\le\delta L(\theta)\}.
\label{main:tube-set}
\end{equation}
The name \emph{tube} refers to a band extending $\delta L(\theta)$ above and below the benchmark moment as $\theta$ varies.
The set $\mathcal T_\delta(Q_0)$ consists of the parameter values where this band reaches zero.
For $\mathcal H=\mathcal P(\mathcal X^\star)$, the exact upper endpoint also has the representation
\begin{equation}
\begin{aligned}
&\max_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[g_\theta(X^\star)]\\
&\quad=\inf_{\lambda\ge0}\left\{\lambda\delta+
\mathbb E_{Z^\star\sim Q_0}\!\left[
\max_{x^\star\in\mathcal X^\star}
\{g_\theta(x^\star)-\lambda d(x^\star,Z^\star)\}
\right]\right\}.
\end{aligned}
\label{main:dual}
\end{equation}
The lower endpoint is the negative of this expression with $g_\theta$ replaced by $-g_\theta$.
\end{theorem}
The tube $\mathcal T_\delta(Q_0)$ needs only $\mathbb E_{Q_0}[g_\theta(X^\star)]$, $\delta$, and $L(\theta)$, so it can be evaluated without optimizing over laws.
It may be wider than the DRO set because some moment values within the band cannot be attained by laws in $\mathcal A_\delta(Q_0)$.
The dual representation in~\eqref{main:dual} replaces infinite-dimensional optimization over probability laws with a scalar convex minimization whose objective involves maximizations over latent states.
Appendix~\ref{app:dual-computation} gives a finite approximation with explicit error bounds.
This representation is an application of established Wasserstein duality \cite{BlanchetMurthy2019,GaoKleywegt2023}.
This dual procedure applies to $\mathcal H=\mathcal P(\mathcal X^\star)$.
Additional retained restrictions must remain in the optimization, as in the finite programs of Appendix~\ref{app:finite-program}.
These calculations are at a fixed $\theta$ and determine whether that parameter is feasible.
The parameter set itself need not be an interval.
\subsection{Calibrating the radius and computing the parameter set}
\label{sec:parameter-computation}
For a given benchmark, we first calibrate its radius and then compute the parameter set.
Algorithm~\ref{alg:main-dro-construction} in Section~\ref{sec:recentering} combines these calculations with benchmark choice through recentering.
\paragraph{Calibrate the radius.}
For a chosen $Q_0\in\mathcal H$, any $\delta\ge r^\star(Q_0)$ in \eqref{eq:covering-radius} preserves every compatible parameter value.
A practical way to obtain such a budget is to construct $Q_0$ by imputing unobserved values with a measurable rule $b_0:\mathcal X\to\mathcal X^\star$, so that $Q_0=(b_0)_\#P_X$.
Couple each compatible latent state with the imputed state from its own observation.
This gives
\begin{equation}
r^\star(Q_0)\le\overline r(b_0)
:=\sup_{Q\in\mathcal Q_f(P_X)}
\mathbb E_Q d\bigl(X^\star,b_0(f(X^\star))\bigr).
\label{eq:calibration-certificate}
\end{equation}
The bound holds because Wasserstein distance is no greater than the cost of this particular coupling.
Computing the supremum can still be difficult.
The operational input is any verified upper bound, which need not equal either $\overline r(b_0)$ or $r^\star(Q_0)$.
For example, if a known, integrable function $h$ satisfies
\[
d\bigl(x^\star,b_0(f(x^\star))\bigr)\le h(f(x^\star))
\quad\text{for every admissible latent state},
\]
then $\delta=\mathbb E_{P_X}h(X)$ is valid without optimizing over compatible laws.
Maintained moment bounds can also supply a certificate.
Below, $(1-D)Y\le1-D$ gives $\delta=p_{00}$ for missing outcomes.
Appendix~\ref{app:recenter-coupling} proves the general rule, including projected states and moment bounds.
The relaxation saves computation only if we can readily obtain a valid covering radius and compute moment bounds over the resulting class.
The computational cost of a linear program also depends on its size and the work needed to construct it.
The network example below illustrates this distinction: summing known link capacities gives the radius, while obtaining the sharp parameter endpoint requires a global search over hidden roles.
Appendix~\ref{app:radius-network} explains why calibration is simpler in this construction.
\paragraph{Compute the parameter set.}
For a calibrated pair $(Q_0,\delta)$, Theorem~\ref{main:feasibility} characterizes the parameter set through the smallest and largest expected moments over $\mathcal A_\delta(Q_0)$.
The examples compute these moment bounds by optimization; analytical formulas provide independent checks of the calculations.
Appendix~\ref{app:computational-analysis} develops numerical methods when explicit calculations are unavailable.
It supplies affine endpoint search and an interval set-inversion guarantee for a fixed calibrated benchmark and radius.
\subsection{Applying the results to the examples}
\label{sec:dro-examples}
For each moment in Section~\ref{sec:model}, we specify the retained laws $\mathcal H$, choose the benchmark $Q_0$, and compute a radius $\delta$ covering the compatible laws.
These choices define $\mathcal A_\delta(Q_0)$.
Theorem~\ref{main:feasibility} then gives the DRO set from its smallest and largest expected moments.
Theorem~\ref{main:tube} gives the comparison bound using the benchmark moment and $L(\theta)$.
Appendices~\ref{app:missingness} and~\ref{app:network-calculation} supply complete derivations and numerical certificates.
\Needspace{7\baselineskip}
\paragraph{Missing outcomes.}
This example shows how the DRO interval can exceed the sharp bounds by allowing changes to outcomes that are already observed.
Section~\ref{sec:recentering-examples} shows how recentering recovers the sharp set.
Keep $g_\theta(D,Y)=\theta-Y$ from Example~\ref{ex:manski-missingness}.
Assign zero to every missing outcome to obtain $Q_0=Q_{t=0}$.
Take $\mathcal H=\mathcal P(\{0,1\}^2)$ and Hamming distance, which counts changed binary coordinates.
The imputation cost $(1-D)Y\le1-D$ directly certifies the radius $p_{00}$.
Here the certificate is exact: a compatible imputation law $Q_t$ changes a fraction $t$ of the missing outcomes from zero to one.
Moving this mass costs $p_{00}t$, so the radius is
\[
\delta=r^\star(Q_0)
=\sup_{t\in[0,1]}W_1(Q_t,Q_0)
=\sup_{t\in[0,1]}p_{00}t=p_{00}.
\]
The moment optimizations are
\[
\begin{aligned}
\min_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[g_\theta(D,Y)]
&=\theta-\min\{1,p_{11}+\delta\},\\
\max_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[g_\theta(D,Y)]
&=\theta-\max\{0,p_{11}-\delta\}.
\end{aligned}
\]
Theorem~\ref{main:feasibility} retains the values for which zero lies between these bounds, giving
\begin{equation}
\Theta_{\mathrm{DRO}}(\delta;Q_0)
=[\max\{0,p_{11}-\delta\},\ \min\{1,p_{11}+\delta\}].
\label{ex:missing-dro}
\end{equation}
For Theorem~\ref{main:tube}, the benchmark moment is $\mathbb E_{Q_0}[g_\theta(D,Y)]=\theta-p_{11}$ and $L(\theta)=1$.
Its inequality $|\theta-p_{11}|\le\delta$, restricted to $\Theta=[0,1]$, gives the same interval.
Substituting $\delta=p_{00}=0.2$ and $p_{11}=0.4$ gives $[0.2,0.6]$, compared with the sharp set $[0.4,0.6]$ from Section~\ref{sec:model}.
The interval extends below the sharp lower bound because the transport ball allows changes to workers' observed outcomes.
All calculations are verified in Appendix~\ref{app:missing-dro-calculation}.
\paragraph{Thresholded networks.}
This example gives computable outer bounds when obtaining the sharp set requires a difficult search over hidden network roles.
Continue Example~\ref{ex:lim-threshold} with bounded outcomes $Y$.
For the relaxation, include the observed record $X$ alongside $G^\star$, so candidate laws describe $(X,G^\star)$.
Retain the known law $P_X$ and the nonnegative, zero-diagonal graph support $\mathcal S=\{A\ge0:A_{ii}=0,\ A\mathbf1=\mathbf1\}$.
The class $\mathcal H$ preserves $P_X$ and allows any graph in $\mathcal S$.
It relaxes the common-role restriction and the requirement that the candidate graph reproduce the report.
The observed network $G$ has one unit-weight outgoing link per row, so imputing zero for every hidden link gives
\[
G^0(G)=G,\qquad Q_0=\mathcal L_{P_X}(X,G).
\]
Here $\mathcal L_{P_X}(X,G)$ is the joint law of $X\sim P_X$ and its zero-imputed latent graph $G$.
Let $d_i=\sum_{j:(i,j)\in E}w_{ij}$ and $d_{\max}=\max_i d_i>0$.
We scale graph distances by known row capacities, so rows with less possible hidden weight receive a smaller allowance.
For $x=(y,t,g)$ and $x'=(y',t',g')$, set
\[
\begin{gathered}
\omega_i=\frac1{10}+\frac{9d_i}{10d_{\max}}
\quad(i=1,\ldots,m),\qquad \omega_h=\omega_s=1,\\
d_{\mathrm{cap}}(A,B)=\max_{i,j}\frac{|A_{ij}-B_{ij}|}{\omega_i},\\
d((x,A),(x',B))=\|y-y'\|_\infty
+10\mathbf1\{(t,g)\ne(t',g')\}+d_{\mathrm{cap}}(A,B).
\end{gathered}
\]
The positive floor keeps the distance well-defined for rows with no eligible links.
The calibrated radius is
\[
\delta=r^\star(Q_0)
=\mathbb E_{P_X}\!\left[\sup_{A\in\Gamma(G)}d_{\mathrm{cap}}(A,G)\right]
=\eta d_{\max}<\min\{\xi,1-\xi\}.
\]
The graph diameter under $d_{\mathrm{cap}}$ is at most $10$, making the same-record coupling optimal for $Q_0$; see Appendix~\ref{app:network-calculation}.
Here $\Gamma(G)$ is the set of latent networks that could have produced $G$ subject to the role and weight restrictions in Example~\ref{ex:lim-threshold}.
The radius requires only sums of known capacities.
It does not require finding the role assignment with the greatest total exposure.
For this benchmark, the exposure optimizations have explicit values:
\[
\begin{aligned}
\underline b_\delta
&:=\min_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[T^\top G^\star T]
=1-\delta,\\
\overline b_\delta
&:=\max_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[T^\top G^\star T]
=1+\delta\sum_{i=1}^m\omega_i.
\end{aligned}
\]
The lower bound allows weight to move from the baseline treated link to the untreated contact.
For the upper bound, core row $i$ can move weight $\delta\omega_i$ toward treated peers.
Since $\mathcal H$ retains the observed law, the moment extrema in Theorem~\ref{main:feasibility}, for $\theta\ge0$, are
\[
\begin{aligned}
\min_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[g_\theta]
&=\mathbb E_{P_X}[T^\top(Y-\beta_0T)]
-\theta\left(1+\delta\sum_{i=1}^m\omega_i\right),\\
\max_{Q\in\mathcal A_\delta(Q_0)}\mathbb E_Q[g_\theta]
&=\mathbb E_{P_X}[T^\top(Y-\beta_0T)]-\theta(1-\delta).
\end{aligned}
\]
For a positive observed contribution, the theorem therefore gives
\begin{equation}
\Theta_{\mathrm{DRO}}(\delta;Q_0)
=\Theta\cap\left[
\frac{\mathbb E_{P_X}[T^\top(Y-\beta_0T)]}{1+\delta\sum_{i=1}^m\omega_i},\,
\frac{\mathbb E_{P_X}[T^\top(Y-\beta_0T)]}{1-\delta}\right].
\label{ex:network-dro}
\end{equation}
Since $\mathcal H$ fixes the observed contribution, we apply Theorem~\ref{main:tube} to the equivalent moment $\mathbb E_{P_X}[T^\top(Y-\beta_0T)]-\theta T^\top G^\star T$.
Its expectation equals that of $g_\theta$ throughout $\mathcal H$.
Exposure is $(1+\sum_{i=1}^m\omega_i)$-Lipschitz, giving the \emph{exposure tube}\footnote{This is the residual tube in Appendix~\ref{app:recentering-network}.
It need not equal \eqref{main:tube-set} computed from the original moment $g_\theta$.}
\[
\left|\mathbb E_{P_X}[T^\top(Y-\beta_0T)]-\theta\right|
\le\delta\left(1+\sum_{i=1}^m\omega_i\right)|\theta|.
\]
The factor adds the scaled allowances of the core rows and the treated contact.
The computational gain is a reusable procedure: calibrate from capacities, optimize moments over the retained envelope, and compute parameter bounds.
Its implementation does not require the analytical endpoint formulas displayed here for validation.
The price is possible excess width, which the capacity comparators in Section~\ref{sec:recentering-examples} help diagnose.
Appendix~\ref{app:network-calculation} proves the formulas and connects sharp computation to maximum cut.
Its 152-node numerical illustration is shown in Figure~\ref{fig:dro-example-sets}.
The regression application in Appendix~\ref{app:regression-calculation} shows how a bound on measurement-error size supplies a valid radius without assuming independent errors.
Retaining the observed joint law and squared-error budget makes moment optimization tractable and recovers the sharp set, with no grid for the true regressor.
\Needspace{3\baselineskip}
The examples separate two sources of excess width: replacing exact compatibility by a transport class, and replacing exact transport optimization by the tube.
This distinction guides the choice of benchmark and radius developed next.
\FloatBarrier
\subsection{Recentering to tighten the outer set}
\label{sec:recentering}
A benchmark obtained by imputation can lie near the edge of the compatible class.
Recentering seeks a benchmark with a smaller covering radius, while keeping the observed law and maintained model fixed.
Coverage and improvement require separate checks: the new envelope must still contain every compatible law, and must lie within the old envelope.
We first give the exact nesting criterion, then a less costly sufficient certificate.
For a candidate $B\in\mathcal H$ and the radius $\delta_B$ actually used, define
\begin{equation}
\Phi(B,\delta_B;Q_0)
:=\max_{\substack{Q\in\mathcal H\\W_1(Q,B)\le\delta_B}}W_1(Q,Q_0).
\label{rec:phi}
\end{equation}
This is the largest distance from the old benchmark among laws allowed by the proposed envelope.
The retained restrictions enter the test and can permit nesting even when the full balls would not be nested.
Changing the benchmark can admit different latent laws even when the covering radius decreases.
The next theorem identifies when recentering preserves coverage and yields a DRO set contained in the initial one.
\begin{theorem}[Coverage and exact envelope nesting]
\label{main:recentering}
Under Assumptions~\ref{ass:ambient-regularity} and \ref{ass:retained-class}, $r^\star$ is convex under mixtures, 1-Lipschitz in its benchmark, and attains a minimum on $\mathcal H$.
For $B,Q_0\in\mathcal H$ and nonnegative radii,
\begin{equation}
\Phi(B,\delta_B;Q_0)\le\delta_0
\quad\Longleftrightarrow\quad
\mathcal A_{\delta_B}(B)\subseteq\mathcal A_{\delta_0}(Q_0).
\label{rec:exact-nesting}
\end{equation}
If these equivalent nesting conditions hold and $\delta_B\ge r^\star(B)$, then
\begin{equation}
\Theta_{\mathrm{ID}}(P_X)
\subseteq\Theta_{\mathrm{DRO}}(\delta_B;B)
\subseteq\Theta_{\mathrm{DRO}}(\delta_0;Q_0).
\label{rec:nested-dro}
\end{equation}
\end{theorem}
\Needspace{5\baselineskip}
Evaluating the exact nesting criterion may require a difficult optimization.
The triangle inequality provides a simpler sufficient certificate, which also guarantees tube nesting under Assumption~\ref{ass:moment-sensitivity}.
The triangle test can reject every benchmark change when the initial radius is exact; Algorithm~\ref{alg:main-dro-construction} therefore also checks sharper nesting certificates (Appendix~\ref{app:three-point}).
\begin{corollary}[Triangle certificate for nested bounds]
\label{main:triangle-certificate}
Under Assumptions~\ref{ass:ambient-regularity} and \ref{ass:retained-class}, for $B,Q_0\in\mathcal H$ and nonnegative radii,
\begin{equation}
\Phi(B,\delta_B;Q_0)\le\delta_B+W_1(B,Q_0)
\label{rec:triangle-upper}
\end{equation}
gives an inexpensive sufficient certificate.
In particular,
\begin{equation}
\delta_B+W_1(B,Q_0)\le\delta_0
\label{rec:nesting-condition}
\end{equation}
together with $\delta_B\ge r^\star(B)$ guarantees \eqref{rec:nested-dro} and, under Assumption~\ref{ass:moment-sensitivity},
\begin{equation}
\Theta_{\mathrm{ID}}(P_X)\subseteq\mathcal T_{\delta_B}(B)
\subseteq\mathcal T_{\delta_0}(Q_0).
\label{rec:nested-tubes}
\end{equation}
For $\delta_0\ge r^\star(Q_0)$, the triangle-certified benchmarks $\{B\in\mathcal H:r^\star(B)+W_1(B,Q_0)\le\delta_0\}$ form a nonempty compact convex set.
Minimizing $r^\star$ on this set is a convex program with an attained minimum, and an LP when the compatible class has finite generators on finite support with linear retained restrictions (Appendix~\ref{app:recenter-finite}).
\end{corollary}
The exact criterion guarantees weak DRO-set inclusion.
A smaller radius need not strictly tighten the parameter set, nor does exact envelope nesting imply nesting of the separately constructed tubes.
For a candidate accepted only by the sharper test, report $\mathcal T_{\delta_0}(Q_0)\cap\mathcal T_{\delta_B}(B)$ when a nested tube refinement is desired.
\paragraph{Benchmark selection.}
Given a declared family $\mathcal B\subseteq\mathcal H$ and a valid coverage certificate $\overline r(B)\ge r^\star(B)$, the sharper ideal program is
\begin{equation}
\min_{B\in\mathcal B}\overline r(B)
\quad\text{subject to}\quad
\Phi(B,\overline r(B);Q_0)\le\delta_0.
\label{rec:center-rule}
\end{equation}
For the same family and radius rule, its feasible set contains every triangle-certified candidate, so its global optimum cannot have a larger radius and can have a strictly smaller one.
General convexity is not asserted.
Appendix~\ref{app:exact-nesting} gives conditions for attainment, and Appendix~\ref{app:three-point} illustrates a strict gain over the triangle rule.
The triangle rule remains useful: computable coverage and movement bounds $\overline r(\alpha)$ and $\overline s(\alpha)$ yield
\begin{equation}
\min_{\alpha\in A}\{\overline r(\alpha):
\overline r(\alpha)+\overline s(\alpha)\le\delta_0\}.
\label{rec:certificate-rule}
\end{equation}
With compact convex $A$, continuous convex certificates, and a feasible baseline, \eqref{rec:certificate-rule} is a convex program with an attained minimum.
Its solution certifies coverage and nested tubes; exact nesting can also certify benchmarks that fail the triangle test.
Algorithm~\ref{alg:main-dro-construction} combines calibration, benchmark selection, and moment optimization into a single procedure.
It verifies the two proposals from the constrained and unrestricted radius programs before computing the final bounds.
Figure~\ref{fig:algorithm-recentering} summarizes the algorithm and the corresponding inclusions of law envelopes and parameter sets.
For a fixed proposal $(B,\delta_B)$, let $U_\Phi\ge\Phi(B,\delta_B;Q_0)$ denote a certified upper bound, including numerical error; use $U_\Phi=+\infty$ when no such certificate is available.
The nesting test evaluates the proposed pair without changing it.
\par\addvspace{\baselineskip}
\begingroup
\setstretch{1.05}
\Needspace{\textheight}
\hrule\vspace{2pt}
\captionsetup{font=normalsize,labelfont=bf,labelsep=space,
justification=raggedright,singlelinecheck=false,skip=2pt,hypcap=false}
\captionof{algorithm}{Constructing and certifying recentered bounds}
\label{alg:main-dro-construction}
\hrule\vspace{3pt}
\textbf{Inputs.}
Measurement map $f:\mathcal X^\star\to\mathcal X$, maintained class $\mathcal C$, observed law $P_X$, moment $g_\theta$, parameter space $\Theta$, metric $d$, and retained class $\mathcal H$ satisfying Assumptions~\ref{ass:ambient-regularity}--\ref{ass:retained-class}; benchmarks $Q_\alpha\in\mathcal H$, $\alpha\in A$, and certified moment routines.
For tubes, also supply $L(\theta)$ as in Assumption~\ref{ass:moment-sensitivity}.
\begin{enumerate}[label=\textbf{\arabic*.},leftmargin=*,itemsep=4pt]
\item \textbf{Choose an initial benchmark and a radius ensuring coverage.}
Choose $Q_0\in\mathcal H$; calibrate by \eqref{eq:calibration-certificate} or \eqref{cal:covering-bound} and initialize
\[
\delta_0\ge r^\star(Q_0),\qquad
(Q_{\mathrm c},\delta_{\mathrm c})\gets(Q_0,\delta_0).
\]
The current pair $(Q_{\mathrm c},\delta_{\mathrm c})$ remains available if no proposal improves it.
\item \textbf{Propose new benchmarks and their covering radii.}
Bound each benchmark's covering radius and distance from $Q_0$:
\[
r^\star(Q_\alpha)\le\overline r(\alpha),\qquad
W_1(Q_\alpha,Q_0)\le\overline s(\alpha).
\]
Under the conditions of \eqref{rec:certificate-rule}, compute
\begin{equation}
\begin{aligned}
\alpha_\triangle&\in\operatorname*{arg\,min}_{\alpha\in A}\{\overline r(\alpha):\overline r(\alpha)+\overline s(\alpha)\le\delta_0\},\\
\alpha_{\rm free}&\in\operatorname*{arg\,min}_{\alpha\in A}\overline r(\alpha).
\end{aligned}
\label{rec:unrestricted-proposal}
\end{equation}
The first program enforces nesting; the second seeks a smaller radius without imposing nesting, which Step~3 must verify.\footnote{
In the finite-support setting of Appendix~\ref{app:recenter-finite}, solve \eqref{app:finite-center-lp} to obtain the unrestricted proposal; adding \eqref{app:finite-center-nest-lp} gives the triangle-certified proposal.}
For each $\alpha\in\{\alpha_\triangle,\alpha_{\rm free}\}$, propose the benchmark $B=Q_\alpha$ with covering radius $\delta_B=\overline r(\alpha)$.
\item \textbf{Check nesting and keep the smallest certified radius} (Theorem~\ref{main:recentering}, Corollary~\ref{main:triangle-certificate}).
Coverage follows from the Step~2 certificate $\delta_B=\overline r(\alpha)\ge r^\star(B)$.
Check nesting relative to the initial pair:
\begin{equation}
\min\{\delta_B+\overline s(\alpha),U_\Phi\}\le\delta_0.
\label{rec:acceptance-test}
\end{equation}
\begin{algorithmic}
\For{each $\alpha\in\{\alpha_\triangle,\alpha_{\rm free}\}$ and its pair $(B,\delta_B)$ from Step~2}
\State $U_\Phi\gets+\infty$; first check the triangle certificate $\delta_B+\overline s(\alpha)\le\delta_0$.
\If{the triangle certificate does not establish nesting ($\delta_B+\overline s(\alpha)>\delta_0$)}
\State Compute a sharper certificate $U_\Phi\ge\Phi(B,\delta_B;Q_0)$ analytically or using:\footnote{
For deterministic network imputations satisfying the restrictions in Appendix~\ref{app:network-nesting}, use the exact LP \eqref{app:eq:network-nesting-lp}.}
\Statex \hspace{\algorithmicindent}\hspace{\algorithmicindent}
(i) point-mass $Q_0$: moment program \eqref{eq:pointmass-phi};
\Statex \hspace{\algorithmicindent}\hspace{\algorithmicindent}
(ii) finite support, linear restrictions: upper relaxation \eqref{eq:mccormick-phi}.
\State Keep $U_\Phi=+\infty$ if no upper certificate is obtained.
\EndIf
\If{\eqref{rec:acceptance-test} holds \textbf{and} $\delta_B<\delta_{\mathrm c}$}
\State $(Q_{\mathrm c},\delta_{\mathrm c})\gets(B,\delta_B)$
\Else
\State Keep $(Q_{\mathrm c},\delta_{\mathrm c})$ unchanged.
\EndIf
\EndFor
\end{algorithmic}
\end{enumerate}
\par\vspace{3pt}\hrule
\newpage
\hrule\vspace{2pt}
\noindent\textbf{Algorithm~\ref*{alg:main-dro-construction}} (continued)\par
\vspace{2pt}\hrule\vspace{3pt}
\begin{enumerate}[label=\textbf{\arabic*.},leftmargin=*,itemsep=4pt,start=4]
\item \textbf{Compute the parameter set for the selected pair} (Theorem~\ref{main:feasibility}).
Minimize and maximize the moment over the selected envelope; retain $\theta$ when zero lies between these bounds, as in \eqref{main:feasibility-test}:
\[
\Theta_{\mathrm{DRO}}(\delta_{\mathrm c};Q_{\mathrm c})
=\left\{\theta\in\Theta:
\min_{Q\in\mathcal A_{\delta_{\mathrm c}}(Q_{\mathrm c})}\mathbb E_Qg_\theta
\le0\le
\max_{Q\in\mathcal A_{\delta_{\mathrm c}}(Q_{\mathrm c})}\mathbb E_Qg_\theta\right\}.
\]
\item \textbf{Compute both tubes and report their intersection} (Theorem~\ref{main:tube}).
Using the initial pair from Step~1 and the selected pair from Step~3, compute
\[
\begin{aligned}
\mathcal T_{\delta_0}(Q_0)
&\gets\{\theta\in\Theta:|\mathbb E_{Q_0}g_\theta|\le\delta_0L(\theta)\},\\
\mathcal T_{\delta_{\mathrm c}}(Q_{\mathrm c})
&\gets\{\theta\in\Theta:|\mathbb E_{Q_{\mathrm c}}g_\theta|\le\delta_{\mathrm c}L(\theta)\},\\
\mathcal T_{\rm out}
&\gets\mathcal T_{\delta_0}(Q_0)\cap\mathcal T_{\delta_{\mathrm c}}(Q_{\mathrm c}).
\end{aligned}
\]
\end{enumerate}
\textbf{Output.}
Return $(Q_{\mathrm c},\delta_{\mathrm c})$, $\Theta_{\mathrm{DRO}}(\delta_{\mathrm c};Q_{\mathrm c})$, and, when computed, $\mathcal T_{\rm out}$.
Both sets contain $\Theta_{\mathrm{ID}}(P_X)$; the DRO set refines the initial set by \eqref{rec:nested-dro}.
\par\vspace{3pt}\hrule
\endgroup
\par\addvspace{\baselineskip}
Choose among Step~3's methods according to the problem's structure.
Exact verification on small supports is also available in Appendix~\ref{app:finite-nesting}.
An upper bound exceeding $\delta_0$ alone is inconclusive; a feasible envelope law with distance from $Q_0$ exceeding $\delta_0$ proves failure of envelope nesting.
The regression application in Appendix~\ref{app:recentering-eiv} illustrates exact acceptance when the triangle bound is inconclusive.
The strict improvement rule retains the current pair on ties.
No global solution of \eqref{rec:center-rule} is required.
Step~4 can use affine endpoint search \eqref{app:affine-computation-endpoints} or certified set inversion under the conditions of Proposition~\ref{app:parameter-computation-guarantee}; at finite tolerances, report the certified enclosures on the chosen search interval.
In Step~5, the intersection equals the new tube whenever the triangle certificate guarantees tube nesting.
A benchmark-specific bound on the moment change can further tighten Step~5's tubes, as in Appendix~\ref{app:recentering-eiv}.
Alternatively, intersect calibrated \emph{law} envelopes and solve Step~4's moment programs over that intersection (see the intersection fallback in Appendix~\ref{app:recenter-finite}).
\begin{figure}[htbp]
\centering
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{\textbf{1. Calibrate}\\$Q_0,\ \delta_0$};
\node[stepbox] (certify) at (5.85,6.08)
{\textbf{2. Propose}\\coverage-certified candidates};
\node[stepbox,draw=algorithmteal,fill=algorithmteal!9]
(recenter) at (9.92,6.08)
{\textbf{3. Verify}\\triangle bound or $U_\Phi$};
\node[stepbox] (compute) at (13.99,6.08)
{\textbf{4--5. Compute}\\DRO and tube sets};
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(\from.east) -- (\to.west);
\node[font=\small\bfseries] at (2.6,5.0) {Parameter values};
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(2.65,2.45) ellipse [x radius=2.70,y radius=2.10];
\node[text=black!70] at (4.82,4.00) {$\Theta$};
\path[fill=black!8,draw=black!55,line width=.9pt]
plot[smooth cycle,tension=.85] coordinates {
(.48,2.18) (.96,3.48) (2.50,3.93) (4.32,3.38)
(4.75,2.19) (3.89,1.05) (2.15,.74) (.76,1.20)
};
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{$\Theta_{\mathrm{DRO}}(\delta_0;Q_0)$};
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plot[smooth cycle,tension=.85] coordinates {
(.79,2.20) (1.04,3.05) (2.22,3.39) (3.77,2.98)
(4.02,2.10) (3.24,1.15) (2.05,1.01) (1.05,1.45)
};
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{$\Theta_{\mathrm{DRO}}(\delta_{\mathrm c};Q_{\mathrm c})$};
\begin{scope}[shift={(2.05,2.13)},rotate=28]
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(0,0) ellipse [x radius=1.10,y radius=.62857];
\end{scope}
\node[text=inkblue] at (2.05,2.13) {$\Theta_{\mathrm{ID}}(P_X)$};
\draw[black!60,{Stealth[length=5pt]}-,line width=.9pt]
(5.65,2.45) -- (9.45,2.45);
\node[text=black!75] at (7.55,3.22) {Theorem~\ref{main:feasibility}};
\node at (7.55,2.85) {$\mathbb E_Q[g_\theta(X^\star)]=0$};
\node[align=center,text=black!70] at (7.55,1.94)
{for some $Q$\\in the envelope};
\node[font=\small\bfseries] at (12.80,5.0) {Latent laws};
\filldraw[fill=black!3,draw=black!40,line width=.7pt]
(9.8,.10) rectangle (15.8,4.70);
\node[anchor=north west,text=black!70] at (10.0,4.48) {$\mathcal H$};
\path[fill=black!8,draw=black!55,line width=.9pt]
(13.35,2.45) circle (2.05);
\node[text=black!70] at (14.20,3.62) {$\mathcal A_{\delta_0}(Q_0)$};
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(12.60,2.45) circle (1.30);
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{$\mathcal A_{\delta_{\mathrm c}}(Q_{\mathrm c})$};
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\end{scope}
\node[text=inkblue] (compatiblelabel) at (10.50,2.47) {$\mathcal Q_f(P_X)$};
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(compatiblelabel.east) -- (11.47,2.47);
\draw[black!55,line width=.75pt] (13.35,2.45) -- (14.695, .903);
\node[fill=black!8,text=black!70,inner sep=1pt]
at (14.20,1.54) {$\delta_0$};
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\node[text=algorithmteal] at (12.33,1.50) {$\delta_{\mathrm c}$};
\draw[algorithmteal,-{Stealth[length=4pt]},line width=.9pt]
(13.35,2.58) .. controls (13.30,3.01) and (12.67,3.01) .. (12.60,2.58);
\node[font=\scriptsize,text=algorithmteal] at (13.04,3.08) {recenter};
\fill[black!70] (13.35,2.45) circle (2pt);
\node[anchor=west,text=black!70] at (13.48,2.47) {$Q_0$};
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\node[anchor=west,text=algorithmteal] at (12.69,2.18) {$Q_{\mathrm c}$};
\node at (12.80,-.32)
{$\Phi(Q_{\mathrm c},\delta_{\mathrm c};Q_0)\le\delta_0$};
\end{tikzpicture}
}
\caption{From calibrated law envelopes to parameter bounds.
Coverage preserves the compatible class in blue; exact nesting keeps the teal envelope inside the gray one.
The triangle condition is a sufficient special case, drawn here schematically.
Moment optimization maps these law inclusions to the parameter inclusions on the left.}
\label{fig:algorithm-recentering}\label{fig:ambiguity-geometry}
\end{figure}
\FloatBarrier
\subsection{Recentering in the examples}
\label{sec:recentering-examples}
Building on Section~\ref{sec:dro-examples}, the examples apply Algorithm~\ref{alg:main-dro-construction} to choose a new benchmark and radius, then recompute the parameter bounds.
\paragraph{Missing outcomes.}
Here recentering removes all excess width and recovers the sharp identified interval.
Keep $g_\theta(D,Y)=\theta-Y$, the zero imputation $Q_0=Q_{t=0}$, and its radius $\delta_0=p_{00}$.
A candidate $Q_t$ leaves observed outcomes unchanged and assigns success probability $t$ to missing outcomes.
Its covering and movement certificates are exact:
\[
\overline r(t)=r^\star(Q_t)=p_{00}\max\{t,1-t\},\qquad
\overline s(t)=W_1(Q_t,Q_0)=p_{00}t.
\]
Equation~\eqref{rec:certificate-rule} becomes
\[
\min_{0\le t\le1}
\left\{p_{00}\max\{t,1-t\}:
p_{00}\max\{t,1-t\}+p_{00}t\le p_{00}\right\}.
\]
The midpoint $t=1/2$ solves this problem, giving
\[
(Q_{\mathrm c},\delta_{\mathrm c})=(Q_{1/2},p_{00}/2),\qquad
\delta_{\mathrm c}+W_1(Q_{\mathrm c},Q_0)=p_{00}=\delta_0.
\]
This benchmark attains the globally smallest covering radius over $\mathcal H$, hence also solves \eqref{rec:center-rule} with exact radii.
The displayed equality verifies the nesting condition \eqref{rec:nesting-condition} in Corollary~\ref{main:triangle-certificate}.
For Theorem~\ref{main:feasibility}, the exact moment extrema are
\[
\begin{aligned}
\min_{Q\in\mathcal A_{\delta_{\mathrm c}}(Q_{\mathrm c})}
\mathbb E_Q[\theta-Y]&=\theta-(p_{11}+p_{00}),\\
\max_{Q\in\mathcal A_{\delta_{\mathrm c}}(Q_{\mathrm c})}
\mathbb E_Q[\theta-Y]&=\theta-p_{11}.
\end{aligned}
\]
They are attained by imputing one and zero for all missing outcomes, respectively.
Requiring zero to lie between these extrema gives
\[
\Theta_{\mathrm{DRO}}(\delta_{\mathrm c};Q_{\mathrm c})
=[p_{11},p_{11}+p_{00}].
\]
The inequality in Theorem~\ref{main:tube}, $|\theta-(p_{11}+p_{00}/2)|\le p_{00}/2$, gives the same interval for the tube.
In the numerical example, recentering raises the lower bound from $0.2$ to $0.4$, while the upper bound remains $0.6$.
Both the DRO interval and the tube therefore have half their original width.
Appendix~\ref{app:recentering-missing} provides details.
\paragraph{Thresholded networks.}
Here recentering tightens the computable bounds without requiring the sharp set to be computed.
Continue with the same observed law, capacity-weighted distance, and initial radius $\delta_0=\eta d_{\max}$.
Let $G_t(G)$ assign the fraction $t\in[0,1/2]$ of each eligible weak-link capacity and reduce its untreated-contact weight to keep the row sum equal to one.
Write $Q_t=\mathcal L_{P_X}(X,G_t(G))$.
The exact covering and movement certificates are
\[
\overline r(t)=r^\star(Q_t)=(1-t)\delta_0,
\qquad \overline s(t)=W_1(Q_t,Q_0)=t\delta_0.
\]
Consequently, \eqref{rec:certificate-rule} becomes
\[
\min_{0\le t\le1/2}
\bigl\{(1-t)\delta_0:
(1-t)\delta_0+t\delta_0\le\delta_0\bigr\}.
\]
Its solution is the midpoint imputation, $Q_{\mathrm c}=Q_{1/2}$, with $\delta_{\mathrm c}=\delta_0/2$.
Corollary~\ref{main:triangle-certificate} applies because the new radius and movement sum to $\delta_0$.
Compatible laws separated by $\delta_0$ show that every benchmark covering them needs radius at least $\delta_0/2$.
Thus this midpoint is already globally optimal for covering radius; the sharper criterion cannot reduce that radius further.
The recentered exposure bounds are
\[
\underline b_{\mathrm c}=1-\delta_0/2,\qquad
\overline b_{\mathrm c}
=1+\frac{\eta}{2}\sum_{(i,j)\in E}w_{ij}
+\frac{\delta_0}{2}\sum_{i=1}^m\omega_i.
\]
The implementation computes these exposure extrema by constrained optimization, supplies them to the moment routine, and searches for the parameter endpoints.
The formulas validate those computations.
For the same positive observed contribution, the resulting interval is
\[
\Theta_{\mathrm{DRO}}(\delta_{\mathrm c};Q_{\mathrm c})
=\Theta\cap\left[
\frac{\mathbb E_{P_X}[T^\top(Y-\beta_0T)]}{\overline b_{\mathrm c}},\,
\frac{\mathbb E_{P_X}[T^\top(Y-\beta_0T)]}{\underline b_{\mathrm c}}
\right].
\]
The positive floor in the metric weights yields $\delta_0\sum_{i=1}^m\omega_i-\eta\sum_e w_e =\eta(m d_{\max}-\sum_e w_e)/10>0$ in this instance.
Both interval endpoints move inward when $\delta_0>0$ and the observed contribution is positive, before intersection with $\Theta$.
Thus recentering strictly reduces the width of this unrestricted interval.
It need not recover the sharp set.
Holding the same observed contribution fixed, Theorem~\ref{main:tube} gives the nested exposure tube
\[
\left|\mathbb E_{P_X}[T^\top(Y-\beta_0T)]
-\theta\left(1+\frac{\eta}{2}\sum_{(i,j)\in E}w_{ij}\right)\right|
\le\frac{\delta_0}{2}\left(1+\sum_{i=1}^m\omega_i\right)|\theta|.
\]
Appendix~\ref{app:recentering-network} proves these calculations and gives the numerical bounds in Figure~\ref{fig:recentering-sets}.
\begin{remark}
Known link capacities can further strengthen $\mathcal H$ within the same moment-optimization framework.
In this example, retaining the capacity relaxation gives tighter bounds than the displayed DRO intervals; both calibrated transport balls then contain the entire retained class, so recentering leaves that bound unchanged.
Inexpensive maximum-cut certificates can strengthen it further (Appendix~\ref{app:network-comparators}).
\end{remark}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,font=\footnotesize,line cap=round]
\node[font=\small\bfseries] at (5.3,.65) {Missing outcomes};
\node[font=\small\bfseries] at (9.5,.65) {Thresholded network};
\node[anchor=east] at (3.25,-0.000) {Sharp};
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\fill[black!65] (6.200000,-0.000) circle (1.6pt);
\fill[black!65] (5.300000,-0.000) circle (1.6pt);
\draw[black!65,line width=2pt] (8.441507,-0.000) -- (9.308511,-0.000);
\fill[black!65] (9.308511,-0.000) circle (1.6pt);
\fill[black!12] (8.199103,-0.090) rectangle (8.441507,0.090);
\draw[black!50,densely dashed,line width=1.3pt] (8.199103,-0.000) -- (8.441507,-0.000);
\node[anchor=east] at (3.25,-0.530) {Initial DRO};
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\fill[black!45] (7.860001,-0.530) circle (1.6pt);
\node[anchor=east] at (3.25,-1.060) {Recentered DRO};
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\fill[inkblue] (5.300000,-1.060) circle (1.6pt);
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\node[anchor=east] at (3.25,-1.590) {Initial tube};
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\fill[orange!40!black] (7.839239,-1.590) circle (1.6pt);
\node[anchor=east] at (3.25,-2.120) {Recentered tube};
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\fill[orange!85!black] (5.300000,-2.120) circle (1.6pt);
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\fill[orange!85!black] (7.950606,-2.120) circle (1.6pt);
\draw[black!55] (3.5,-2.54) -- (7.1,-2.54);
\draw[black!55] (3.500000,-2.54) -- (3.500000,-2.61);
\node[anchor=north,font=\scriptsize] at (3.500000,-2.66) {0};
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\node[anchor=north,font=\scriptsize] at (4.400000,-2.66) {0.2};
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\node[anchor=north,font=\scriptsize] at (5.300000,-2.66) {0.4};
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\node[anchor=north,font=\scriptsize] at (6.200000,-2.66) {0.6};
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\node[anchor=north,font=\scriptsize] at (7.100000,-2.66) {0.8};
\draw[black!55] (7.7,-2.54) -- (11.3,-2.54);
\draw[black!55] (7.776596,-2.54) -- (7.776596,-2.61);
\node[anchor=north,font=\scriptsize] at (7.776596,-2.66) {0.9};
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\node[anchor=north,font=\scriptsize] at (8.542553,-2.66) {0.95};
\draw[black!55] (9.308511,-2.54) -- (9.308511,-2.61);
\node[anchor=north,font=\scriptsize] at (9.308511,-2.66) {1};
\draw[black!55] (10.074468,-2.54) -- (10.074468,-2.61);
\node[anchor=north,font=\scriptsize] at (10.074468,-2.66) {1.05};
\draw[black!55] (10.840426,-2.54) -- (10.840426,-2.61);
\node[anchor=north,font=\scriptsize] at (10.840426,-2.66) {1.1};
\node at (5.3,-3.36) {$\theta$};
\node at (9.5,-3.36) {$\theta$};
\end{tikzpicture}
\caption{Population bounds before and after recentering.
The network intervals bound the moment-defined exposure slope in \eqref{eq:network-moment}.
The network tubes hold the observed contribution fixed before applying Theorem~\ref{main:tube}.
The missing-outcome sharp interval is known exactly.
For networks, the solid sharp segment is a certified inner interval and the dashed segment brackets the unknown lower endpoint.}
\label{fig:recentering-sets}\label{fig:dro-example-sets}
\end{figure}
\FloatBarrier
Recentering can remove excess width from an outer approximation, but it cannot reduce the sharp identified set without additional information or assumptions.
\paragraph{Bounding the unobserved component.}
\label{disc:benchmark-residual}
When part of a moment is determined by the observed law, we can hold its expectation fixed and bound only the remaining residual.
Step~4 of Algorithm~\ref{alg:main-dro-construction} can optimize the residual over its calibrated envelope, as in the regression application.
Further bounds from the maintained restrictions can also be intersected with the DRO set while preserving every compatible value.
Appendix~\ref{app:population-residuals} proves the residual identity; Appendices~\ref{app:network} and~\ref{app:measurement-error} give its applications.
\clearpage
\section{Discussion and conclusion}
\label{sec:conclusion}
We asked how to compute and refine parameter bounds from imperfect measurements while preserving every value allowed by the data and assumptions.
Our framework combines measurement restrictions, moment optimization, and benchmark selection to refine outer bounds while holding the observed law and identifying assumptions fixed.
Algorithm~\ref{alg:main-dro-construction} implements our main contribution: a reusable methodology for calibration, benchmark selection, and parameter-bound computation.
The theory ensures that accepted benchmark changes preserve every compatible parameter value without widening the initial bounds.
The network application shows how known weight limits and benchmark selection yield tighter bounds on the moment-defined network-exposure slope without solving the difficult sharp identification problem or strengthening assumptions.
Recentering recovers the sharp interval for missing outcomes.
In regression, retaining the observed joint law and measurement-error budget makes the optimized bounds sharp.
Future work can develop methods for estimating the bounds and constructing confidence sets for the parameter and its identified set, accounting for sampling uncertainty and benchmark selection.
It can also clarify how imperfect measurement affects attainable estimation rates and the precision of inference.
\clearpage
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\phantomsection
\addcontentsline{toc}{section}{References}
\putbib
}
\end{bibunit}
\clearpage