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Quantile-Only Regression: Inference and Partial Identification under Quantile Disclosure
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\title{\bf Quantile-Only Regression: Inference and Partial Identification under Quantile Disclosure}
\author{Masamune Iwasawa\hspace{.2cm}\\
Doshisha University, 602-8580 Karasuma-higashi-iru, Imadegawa-dori,\\ Kamigyo-ku, Kyoto-shi, Email: [email removed]\\
and \\
Hayato Nakanishi \\
Kanagawa University, 3-27-1 Rokkakubashi Kanagawa-ku, Yokohama,\\
Kanagawa 221-8686, Japan,
Email: [email removed]}
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{\LARGE\bf Title}
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\begin{abstract}
Researchers often observe group quantiles but not group means, raising the question of whether these summaries alone support conventional regression analysis. We develop estimation and inference methods for linear models of group means under quantile-only and mixed disclosure. We establish conditions on disclosure design for consistent estimation and asymptotic normality at the parametric rate. Under fixed disclosure, we characterize sharp identified sets and construct simultaneous confidence regions, clarifying how informativeness depends on the number and placement of disclosed quantiles. Applications to education, social care, and medical expenditure illustrate the methods. The R package \texttt{qonlyreg} implements the proposed procedures.
\end{abstract}
\noindent
{\it Keywords:}
Group-level summary statistics,
Quantiles,
Partial identification,
Simultaneous confidence regions,
Disclosure design.
\noindent
{\it JEL Classification:}
C13, C18, C81.
\vfill
\newpage
\clubpenalty=10000 \widowpenalty=10000 \displaywidowpenalty=10000
\section{Introduction}\label{sec_intro}
Researchers increasingly encounter data in which outcome distributions are disclosed only through group-level summary statistics rather than individual-level observations.
Official statistical publications, administrative reporting systems, and empirical studies used in meta-analysis often report group sizes and selected quantiles such as medians, quartiles, or deciles, while the underlying microdata and sometimes even the group means are unavailable.
Such disclosure practices arise for a variety of reasons, including privacy, confidentiality, and reporting conventions, and occur in substantively different settings ranging from education and public administration to medical research.
This creates a simple but consequential problem for regression analysis.
Researchers may wish to relate group-average outcomes to observable group characteristics, yet the dependent variable in such a regression---the group mean---is not observed.
For example, substituting reported medians for unobserved group means need not recover the regression coefficients of interest, because differences between group means and medians may vary systematically with group characteristics.
More generally, the disclosed quantiles contain information about the mean, but a finite collection need not determine it exactly.
It is therefore not immediate whether conventional regression analysis remains possible from the disclosed summaries alone.
This paper asks three questions.
First, can regression coefficients defined through group means be consistently estimated when only group-specific quantiles are observed?
Second, how do the number and placement of disclosed quantiles determine the accuracy of estimation and the feasibility of conventional inference?
Third, when the available quantile information is insufficient to point identify the regression coefficients, what can still be learned about them?
We study these questions in a linear model for group-level conditional means.
For each group, the researcher observes the group size, group-level covariates, and a collection of sample quantiles, but not the underlying individual-level outcomes.
We reconstruct the group mean by integrating a piecewise-linear interpolation of the disclosed quantiles and estimate the regression coefficients by a sample-size-weighted minimum-distance procedure.
The framework also accommodates mixed disclosure, in which sample means are directly reported for some groups while only quantile summaries are available for others.
Our results deliver three main messages.
First, sufficiently rich quantile disclosure can support conventional regression analysis.
Under increasing quantile disclosure, the error in reconstructing a group mean can be decomposed into a discretization component arising from the finite quantile grid and a sampling component arising from estimation of the disclosed quantiles.
As the quantile grid becomes denser, the discretization error vanishes, yielding consistent estimation of the regression coefficients. If the grid becomes sufficiently dense relative to the sample size, the resulting estimator attains the usual parametric rate and is asymptotically normal, allowing feasible inference based only on the disclosed summaries.
Importantly, refining the quantile grid reduces the discretization error without worsening the sampling-error rate in mean reconstruction.
Second, fixed quantile disclosure creates an identification problem, not merely a loss of precision.
With a fixed finite set of disclosed quantiles, even the population quantiles generally do not determine the group means.
Distinct conditional distributions can generate exactly the same disclosed population quantiles while implying different group means and regression coefficients.
Consequently, increasing the sample size alone cannot resolve the problem: non-identification persists even when sampling uncertainty vanishes.
For this setting, we derive sharp bounds for the group means, characterize the sharp identified set for the regression coefficients, and construct asymptotically valid simultaneous confidence regions.
Third, the informativeness of quantile disclosure depends on where quantiles are reported, not only on how many are reported.
For point estimation, the discretization-error bound is governed by the maximal spacing of the disclosed quantile grid.
Among designs with the same number of disclosed quantiles, evenly spaced grids minimize this maximal spacing and therefore minimize the grid-dependent component of the error bound.
The implication for partial identification is different.
The width of the identified interval for a group mean depends not only on grid spacing but also on the local variation of the quantile function.
A grid that is favorable for point estimation therefore need not be equally favorable for partial identification, and allocating more quantiles to regions where the quantile function changes rapidly may yield tighter identified intervals.
Taken together, these results show that disclosure design is itself a statistical design problem. The number and placement of disclosed quantiles determine how informative the released summaries are for recovering group means, estimating regression coefficients, and conducting statistical inference.
The framework therefore provides a way to assess not only whether regression analysis is possible from a given statistical release, but also how the design of that release affects estimation and identification.
The proposed methods are straightforward to implement using only the disclosed summaries. The reconstructed group means are available in closed form, and the accompanying R package, \texttt{qonlyreg}, implements the main estimation and inference procedures developed in the paper.\footnote{
The package is available at \url{https://github.com/masaiwasawa/qonlyreg}.
}
We examine the finite-sample performance of the procedures through Monte Carlo simulations and illustrate their scope in three substantively different empirical settings.
The first application uses publicly released National Assessment of Educational Progress (NAEP) mathematics summaries for groups defined by U.S. jurisdiction and economically disadvantaged status.
The point estimates are close to the benchmark based on published group means, and the projected confidence interval narrows substantially when disclosure expands from quartiles to five percentiles.
The second uses publicly released Children in Need assessment-duration summaries from English local authorities, where assessment counts and quartiles are observed but the corresponding means are not. We use a fixed-window comparison motivated by regression discontinuity (RD) to examine differences in mean assessment duration around the 50 percent Labour seat-share threshold.
The point estimates are negative and have small conventional standard errors, but the projected confidence intervals accounting for identification uncertainty are wide and include zero under the support restrictions considered.
The third considers a medical-expenditure meta-analysis with mixed disclosure, in which some study comparisons report sample means while others report only quartiles.
Together, these applications illustrate that the same problem arises in official statistics, administrative data, and published study summaries, and that the framework can accommodate both quantile-only and mixed-disclosure environments.
A related literature studies regression and other statistical relationships when only aggregate rather than individual-level data are available, including work on ecological inference \citep{king97} and regression with grouped or aggregated data (e.g., \citealp{farebrother79,nicoletti12}).
These approaches typically seek to recover individual-level relationships or model parameters from aggregate observations.
Related work exploits aggregate sufficient statistics or moments to recover parameters of richer individual-level models.
\citet{kovalchik13}, for example, recovers parameters of an individual-participant random-effects model from aggregate study-level data, while \citet{mcculloch25} use tract-level means and within-tract second moments together with a small-variance approximation to recover parcel-level treatment-effect parameters.
Our problem is distinct: the target is a regression model for group means, while the group means themselves are unavailable and must be reconstructed or bounded from disclosed quantiles.
The literature on recovering distributional features from grouped or quantile-based summaries includes methods for estimating a single population mean from reported quantiles, primarily in the context of medical meta-analysis \citep{hozo05,wan14,luo18,mcgrath20,cai21}.
These methods are designed for a small number of pre-specified summaries such as medians, ranges, and quartiles.
Our focus is different.
Rather than recovering a single mean, we study regression coefficients defined through multiple group means and characterize how the number and placement of disclosed quantiles affect estimation, inference, and identification.
The paper is also related to the literature on partial identification under incomplete information; see \citet{manski03} and \citet{molinari20} for textbook and survey treatments.
Unlike settings in which identification failure arises from missing or interval-censored outcomes, the non-identification studied here is generated by the disclosure mechanism itself.
An early example in this spirit is \citet{cowell82}, who derive lower and upper bounds for income inequality measures from grouped income data and propose interpolation-based point estimators.
Their analysis combines bounding and interpolation arguments to study functionals of partially observed distributions.
To the best of our knowledge, however, the implications of quantile disclosure for regression identification and inference have received little attention.
Finally, the asymptotic analysis builds on classical results for sample quantiles.
For a fixed quantile index, sample quantiles converge at the usual square-root sample-size rate under standard regularity conditions and admit Bahadur representations; see \citet{bahadur66} and \citet{kiefer67}.
The increasing-disclosure setting considered here raises an additional issue because the number of disclosed quantiles may grow with the sample size and the extreme disclosed quantiles may approach the boundaries of the unit interval.
We therefore establish uniform error bounds that remain valid over such expanding grids and use them to derive the asymptotic behavior of the reconstructed-mean regression estimator.
The remainder of the paper is organized as follows.
Section~\ref{sec_model_estimation} introduces the framework and estimator.
Sections~\ref{sec_regression} and~\ref{sec_fixedDC} develop the theoretical results under increasing and fixed quantile disclosure.
Sections~\ref{sec_simulation} and~\ref{sec_empirical} present simulation and empirical illustrations.
Section~\ref{sec_conclude} concludes.
\section{Model and Estimation}\label{sec_model_estimation}
Let $(Y_i,\bm X_i)$ be independent and identically distributed (i.i.d.) copies of
$(Y,\bm X)$, where $Y_i\in\mathbb R$ is the outcome and
$\bm X_i\in\mathbb R^p$ is an underlying individual-level covariate vector.
The support $\mathcal X$ of $\bm X$ is partitioned into a finite collection of
measurable groups $\{\mathcal G_g\}_{g=1}^G$ satisfying
\[
\mathcal G_g\cap\mathcal G_h=\varnothing\quad(g\ne h),
\qquad \bigcup_{g=1}^G\mathcal G_g=\mathcal X.
\]
Write $\bm X_i\in g$ for $\bm X_i\in\mathcal G_g$.
Each observation belongs to exactly one group, so the total number of underlying individual observations is
$N=\sum_{g=1}^G n_g$, where $n_g$ is the group size.
For each group, define the conditional distribution and quantile functions by
\[
F_g(y)=P(Y\le y\mid\bm X\in g),
\qquad
Q_g(\tau)=\inf\{y\in\mathbb R:F_g(y)\ge\tau\},
\quad \tau\in(0,1).
\]
The researcher observes only the released summaries
\[
\{n_g,\bm X_g,\widehat Q_g(\tau_1),\ldots,
\widehat Q_g(\tau_K)\}_{g=1}^G,
\qquad 0<\tau_1<\cdots<\tau_K<1,
\]
and not the individual-level observations $(Y_i,\bm X_i)$.
The partition is formed by the data provider and treated as given.
The group characteristics $\bm X_g$ are nonrandom and conceptually distinct
from $\bm X_i$: they may be aggregated or externally provided and need not
determine the partition.
Our target is a linear restriction on the group-level conditional means,
\[
m_g=E(Y\mid\bm X\in g)=\bm X_g'\bm\beta_0.
\]
We focus on lower-dimensional, nonsaturated specifications that impose
cross-group restrictions, excluding a full set of group indicators.
This setting includes models obtained by aggregating an individual-level
linear conditional mean specification, \(E(Y\mid \bm X)=\bm X'\bm\beta_0\);
Supplemental Material~G explains the connection.
The baseline setup above focuses on quantile-only disclosure.
The framework also accommodates mixed disclosure, in which sample means
are directly reported for some groups while only quantile summaries are
available for others.
The following examples relate the group-mean model to the applications
in Section~\ref{sec_empirical}.
\begin{example}[Educational achievement]
Let $g$ index groups defined by area $s_g$ and socioeconomic
status $D_g\in\{0,1\}$.
Suppose that the sample size $n_g$ and selected sample quantiles of
test scores are disclosed for each group, but the corresponding
sample mean is not.
We model the population mean test score for group $g$ as
$m_g=\alpha_{s_g}+\beta_0 D_g$, allowing area-specific intercepts
and imposing a common mean score difference between the two status groups.
\end{example}
\begin{example}[Administrative duration outcomes]
Let $g$ index groups defined by local authority and reporting period.
Suppose that the number of completed cases $n_g$ and selected sample
quantiles of case duration are disclosed for each group, but the
corresponding sample mean is not.
Within a fixed window of half-width $h$ around a threshold in an
observed authority characteristic, we model the population mean duration
as $m_g=\alpha_h+\delta_h D_g$, where $D_g\in\{0,1\}$ indicates whether
the characteristic is at or above the threshold.
The coefficient $\delta_h$ is the mean-duration difference between
the two sides within this window.
\end{example}
\begin{example}[Meta-analysis with mixed disclosure]
Let $g$ index the population groups in a meta-analysis, with $j_g$
denoting the study comparison to which group $g$ belongs and
$D_g\in\{0,1\}$ distinguishing the two populations being compared.
Suppose that sample sizes $n_g$ are disclosed for all groups,
while some groups report sample means and others only selected
sample quantiles.
We model the population mean outcome as
$m_g=\alpha_{j_g}+\beta_0 D_g$, allowing comparison-specific intercepts.
The coefficient $\beta_0$ is the mean difference between the $D_g=1$
and $D_g=0$ populations within each comparison.
\end{example}
When $E(|Y|\mid\bm X\in g)<\infty$,
$m_g=\int_0^1 Q_g(\tau)\,d\tau$.
We reconstruct the unobserved quantile function by piecewise linear
interpolation: for $\tau\in[\tau_k,\tau_{k+1}]$,
\[
\widetilde Q_g(\tau)
=\widehat Q_g(\tau_k)
+\frac{\tau-\tau_k}{\tau_{k+1}-\tau_k}
\{\widehat Q_g(\tau_{k+1})-\widehat Q_g(\tau_k)\}.
\]
Below $\tau_1$ and above $\tau_K$, use the constant values
$\widehat Q_g(\tau_1)$ and $\widehat Q_g(\tau_K)$, respectively.\footnote{
A known support endpoint can be included as a deterministic interpolation
node. Such endpoints do not contribute to sampling variance, and the
results below continue to hold with this modification.
}
The reconstructed mean is
\[
\hat m_g=\int_0^1\widetilde Q_g(\tau)\,d\tau.
\]
For example, with only quartiles disclosed, the baseline constant-tail rule gives
\[
\hat m_g
=\frac38\widehat Q_g(0.25)
+\frac14\widehat Q_g(0.50)
+\frac38\widehat Q_g(0.75).
\]
We estimate the regression coefficients by
\[
\hat{\bm\beta}
=\arg\min_{\bm\beta}\sum_{g=1}^G
n_g(\hat m_g-\bm X_g'\bm\beta)^2
=\left(\sum_{g=1}^G n_g\bm X_g\bm X_g'\right)^{-1}
\sum_{g=1}^G n_g\bm X_g\hat m_g.
\]
Piecewise linear interpolation preserves monotonicity, requires no tuning
parameters, and yields a closed-form reconstructed mean.
We use sample-size weights in the minimum-distance estimator to
account for differences in group size without requiring preliminary
variance estimates for point estimation.
To keep the paper focused on how quantile disclosure affects
estimation and inference, we leave the analysis of optimal
weighting to future research.
For mixed disclosure, replace $\hat m_g$ by the reported sample mean
where available.
The theory below focuses on quantile-only disclosure, while
Section~\ref{sec_empirical} illustrates the mixed-disclosure
implementation through a meta-analysis of medical expenditure.
\section{Asymptotic Theory: Increasing Quantile Disclosure}\label{sec_regression}
Throughout this section, $G$ is fixed, each $n_g\to\infty$, and
$K=K_N$ increases with $N=\sum_g n_g$.
We use a common quantile grid across groups to simplify notation
and exposition in the theoretical analysis.
The proposed estimator can also be implemented with
group-specific quantile grids.
Fixed disclosure is studied in Section~\ref{sec_fixedDC}.
\subsection{Assumptions}
\begin{assumption}[Sampling]
\label{ass_sample}
Let $(Y_i, \bm{X}_i)$ be i.i.d.\ random variables.
\end{assumption}
\begin{assumption}[Group-Level Linear Mean Restriction]
\label{ass_linear}
For each $g=1,\dots,G$,
$
m_g = E(Y \mid \bm{X} \in g) = \bm{X}_g'\bm{\beta }_0.
$
\end{assumption}
\begin{assumption}[Quantile Function Smoothness]
\label{ass_smooth}
For each group $g$, the quantile function $Q_g(\tau)$ is nondecreasing and satisfies either:
\begin{itemize}
\item[(i)] Lipschitz continuity: $|Q_g(\tau) - Q_g(\tau')| \le L |\tau - \tau'|$, or
\item[(ii)] Twice continuously differentiable with bounded second derivative:
$
\sup_{\tau\in(0,1)} |Q_g''(\tau)|
\le
M_g
$
for some finite constant $M_g$.
\end{itemize}
\end{assumption}
\begin{assumption}[Quantile Estimation]
\label{ass_quantile}
For each $g=1,\dots,G$,
\[
\sum_{k=1}^{K_N}
w_{k,N}
\bigl|
\widehat Q_g(\tau_k)-Q_g(\tau_k)
\bigr|
=
O_p(r_{g,N}),
\]
where $r_{g,N}\to 0$, and $\{w_{k,N}\}_{k=1}^{K_N}$ are the reconstruction weights defined in the proof of Lemma~\ref{lem_mean}.
\end{assumption}
\begin{assumption}[Design]
\label{ass_design}
For each \(g=1,\ldots,G\),
$
\frac{n_g}{N}
\overset{p}{\rightarrow}
\pi_g
=
P(\bm X_i\in\mathcal G_g)
\in(0,1),
$
and \(\|\bm X_g\|<\infty\).
Moreover,
$
\bm\Sigma_X
=
\sum_{g=1}^G
\pi_g
\bm X_g\bm X_g'
$
is positive definite.
\end{assumption}
Given the population group means, Assumptions~\ref{ass_linear} and
\ref{ass_design} imply that $\bm\beta_0$ uniquely minimizes
$\sum_g\pi_g(m_g-\bm X_g'\bm\beta)^2$.
The proposed estimator minimizes the sample analogue of this
criterion and is well-defined with probability approaching one
under Assumption~\ref{ass_design}.
Primitive conditions for the quantile-estimation error in
Assumption~\ref{ass_quantile} are given in Lemma~\ref{lem_uniform_quantile}.
Both alternatives in Assumption~\ref{ass_smooth} imply that the
outcome distribution within each group has bounded support,
although the endpoints need not be known for the estimation
and inference procedures developed in this section.
The Gaussian and Laplace simulations assess robustness outside these
sufficient conditions; see Section~\ref{sec_simulation} and Supplemental
Material~D.
\subsection{Reconstruction Error and Consistency}
Let $\mathcal T_{K_N}=\{\tau_1,\ldots,\tau_{K_N}\}$ denote the \emph{disclosure design},
with $\tau_0=0$, $\tau_{K_N+1}=1$, and maximal spacing
\[
\Delta(\mathcal T_{K_N})
=\max_{0\le k\le K_N}(\tau_{k+1}-\tau_k).
\]
This includes both boundary gaps. We consider designs satisfying
$\Delta(\mathcal T_{K_N})\to0$.
\begin{lemma}[Mean Reconstruction Error]
\label{lem_mean}
Under Assumptions \ref{ass_smooth} and \ref{ass_quantile}, and $\min_g n_g \to \infty$,
the reconstruction error of the group means satisfies
\[
\max_{1\le g\le G} |\hat m_g-m_g|
=
O_p\!\left(
\Delta(\mathcal T_{K_N})^a
+
r_N
\right),
\]
where $r_N = \max_{1\le g\le G} r_{g,N}$, and
$a=1$ under Assumption \ref{ass_smooth}(i) and $a=2$ under Assumption \ref{ass_smooth}(ii).
\end{lemma}
Lemma~\ref{lem_mean} separates the reconstruction error into two components.
The term $\Delta(\mathcal T_{K_N})^a$ corresponds to the
\emph{discretization error} arising from the finite quantile grid,
while $r_N$ summarizes the contribution of \emph{sampling error}
in the disclosed quantiles.
The discretization bound improves from order
$\Delta(\mathcal T_{K_N})$ under Lipschitz continuity to order
$\Delta(\mathcal T_{K_N})^2$ under the stronger smoothness
condition in Assumption~\ref{ass_smooth}(ii).
For the latter case, integrating the linear-interpolation error
over the intervals between disclosed quantiles yields the
quadratic bound \citep[Chapter~3]{atkinson89}.
On the two boundary intervals, this assumption also implies
bounded $Q_g'$, so the integrated error from constant extrapolation
is bounded by a constant times the square of each interval's length.
The following theorem extends this error bound to the proposed
regression estimator.
\begin{theorem}[Regression Error Decomposition]
\label{thm_main}
Under Assumptions \ref{ass_linear}, \ref{ass_smooth}, \ref{ass_quantile}, and \ref{ass_design},
\[
\hat{\bm{\beta}} - \bm{\beta }_0
=
O_p\!\left(
\Delta(\mathcal{T}_{K_N})^a
+
r_N
\right),
\]
where $a=1$ under Assumption~\ref{ass_smooth}(i) and $a=2$ under Assumption~\ref{ass_smooth}(ii).
\end{theorem}
Theorem~\ref{thm_main} implies that
$\hat{\bm\beta}\xrightarrow{p}\bm\beta_0$
if $\Delta(\mathcal T_{K_N})\to0$ and $r_N\to0$.
The discretization bound depends on the largest gap between
successive quantile indices, including the gaps to zero and one.
Increasing $K_N$ alone does not ensure that this spacing vanishes,
since additional quantiles may be concentrated in only part
of the unit interval.
\subsection{Sufficient Conditions for \texorpdfstring{$\sqrt{N}$}{root-N} Consistency}
\label{subsec_rootN}
Theorem~\ref{thm_main} gives sufficient conditions
$\Delta(\mathcal T_{K_N})^a=O(N^{-1/2})$ and $r_N=O(N^{-1/2})$
for $\sqrt N$-consistency.
The latter follows from a uniform quantile-error bound because the
reconstruction weights are nonnegative and sum to one.
Classical results for fixed interior quantiles \citep{bahadur66,kiefer67}
do not directly cover grids with $\tau_1\to0$ and $\tau_{K_N}\to1$.
The next lemma supplies primitive conditions that allow this boundary
behavior.\footnote{
The asymptotic analysis in this section focuses on continuously distributed outcomes.
Discrete outcomes may have flat or discontinuous quantile functions and require separate treatment.
}
Here, sample quantiles are generalized inverses of the group empirical
distribution:
\[
\widehat Q_g(\tau)=\inf\{y\in\mathbb R:F_{n,g}(y)\ge\tau\},
\qquad
F_{n,g}(y)=\frac{1}{n_g}\sum_{i:\bm X_i\in g}1\{Y_i\le y\}.
\]
\begin{lemma}[A sufficient condition for $\sqrt{n_g}$ quantile reconstruction]
\label{lem_uniform_quantile}
Fix a group $g$.
Suppose
Assumption \ref{ass_sample} and that:
(i) $F_g$ is strictly increasing on its support and admits a density \(f_g\) that is continuously differentiable on its support.
(ii) The support of $F_g$ is a compact interval $[\underline y_g,\overline y_g]$.
(iii) There exists a constant $c_g>0$ such that
$
\inf_{y\in[\underline y_g,\overline y_g]} f_g(y) \ge c_g.
$
Then
\begin{equation}
\label{eq_uniform_bound}
\sup_{1\le k\le K_N}
\bigl|
\widehat Q_g(\tau_k)-Q_g(\tau_k)
\bigr|
=
O_p(n_g^{-1/2}).
\end{equation}
\end{lemma}
Under Assumption~\ref{ass_design}, $n_g^{-1/2}=O_p(N^{-1/2})$, so
\eqref{eq_uniform_bound} implies Assumption~\ref{ass_quantile} with
$r_N=N^{-1/2}$.
No additional rate restriction on $\tau_1$ or $1-\tau_{K_N}$
is needed for this uniform absolute-error bound.
Refining the grid therefore does not worsen the sampling-error
rate in mean reconstruction or in the proposed regression estimator.
Combining Lemma~\ref{lem_uniform_quantile} with
Theorem~\ref{thm_main} yields the following sufficient conditions for $\sqrt{N}$-consistency.
\begin{proposition}[Sufficient conditions for $\sqrt{N}$-consistency]
\label{prop_rootN}
Suppose Assumptions~\ref{ass_linear}, \ref{ass_smooth}, and \ref{ass_design} hold, and that the conditions of Lemma~\ref{lem_uniform_quantile} are satisfied for each $g=1,\ldots,G$.
If the quantile grid satisfies
$
\Delta(\mathcal T_{K_N})^a = O(N^{-1/2}),
$
then
\[
\hat{\bm{\beta}}-\bm{\beta}_0
=
O_p(N^{-1/2}),
\]
where $a=1$ under Assumption~\ref{ass_smooth}(i) and $a=2$ under Assumption~\ref{ass_smooth}(ii).
\end{proposition}
For the evenly spaced grid $\tau_k=k/(K_N+1)$,
$\Delta(\mathcal T_{K_N})=(K_N+1)^{-1}$.
Thus, a sufficient condition is that $K_N$ grow at least on the order
of $N^{1/(2a)}$.
Because $r_N=O(N^{-1/2})$ under the lemma's conditions regardless of
the design, minimizing maximal spacing also minimizes the
grid-dependent component of the error bound.
\begin{corollary}\label{cor_optimal_design}
Under the conditions of Lemma~\ref{lem_uniform_quantile},
the evenly spaced grid minimizes the upper bound in Theorem~\ref{thm_main} by minimizing \(\Delta(\mathcal T_{K_N})\) among all disclosure designs of size \(K_N\).
\end{corollary}
\subsection{Asymptotic Expansion and Normality}\label{subsec_expansion}
To derive the asymptotic distribution of the proposed regression
estimator, we first establish an expansion of its estimation
error that separates discretization bias from sampling variation.
We impose Assumption~\ref{ass_smooth}(ii) throughout this subsection and Section~\ref{subsec_variance}.
Let $b_{g,N}$ denote group $g$'s discretization bias arising from
approximating the quantile function on a finite grid, and let
$\psi_{g,N}$ denote the influence function of the reconstructed mean $\hat m_g$.
Their formulas are given in the proof in Supplemental Material~A.
\begin{lemma}[Asymptotic expansion]
\label{lem_expansion}
Under the conditions of Proposition~\ref{prop_rootN} with Assumption \ref{ass_smooth}(ii),
\[
\hat{\bm{\beta}}-\bm{\beta}_0
=
\bm{B}_N
+
\frac{1}{N}\sum_{g=1}^G \sum_{i\in \mathcal{I}_g}
\bm{\Psi}_{g, N}(Y_i)
+
o_p(N^{-1/2}),
\]
where
$
\bm{B}_N
=
\bm\Sigma_X^{-1}
\sum_{g=1}^G \pi_g \bm X_g b_{g,N}
=
O(\Delta(\mathcal T_{K_N})^2)
$ is a deterministic bias term,
$
\bm{\Psi}_{g, N}(y)
=
\bm\Sigma_X^{-1}\bm X_g\psi_{g,N}(y)
$
is an influence function with
$
E[
\bm\Psi_{g,N}(Y_i)
\mid
\bm X_i\in g
]
=
\bm 0
$.
Moreover, $\bm{\Psi}_{g,N}(Y_i)$ is uniformly bounded in $N$ and hence has finite second moment.
Finally, $\mathcal{I}_g = \{i: \bm X_i \in g\}$ denotes the index set of observations in group $g$.
\end{lemma}
Lemma~\ref{lem_expansion} relies on the uniform Bahadur
representation established in Lemma~\ref{lem_bahadur_grid}.
Because the disclosed grid may approach the boundaries of the
unit interval, standard uniform Bahadur results on fixed compact
subsets of $(0,1)$ do not directly apply.
Under the maintained assumptions, Lemma~\ref{lem_bahadur_grid}
establishes an $o_p(n_g^{-1/2})$ bound on the linearization
remainder uniformly over the entire disclosed grid.
This uniform control allows us to aggregate quantile estimation
errors in mean reconstruction and derive the regression expansion.
Supplemental Material~C presents a standard fixed-interior
representation and its proof for comparison.
Although $\sqrt{N}$-consistency holds, the discretization bias
$\bm B_N$ may remain of first-order importance when
$\Delta(\mathcal T_{K_N})^2$ is of order $N^{-1/2}$.
Centering by this bias gives the following limit.
\begin{theorem}[Bias-corrected asymptotic normality]
\label{thm_bias_clt}
Suppose the conditions of Lemma \ref{lem_expansion} hold.
Then
\[
\sqrt{N}
\bigl(
\hat{\bm{\beta}}-\bm{\beta}_0-\bm B_N
\bigr)
\overset{d}{\longrightarrow}
\mathcal N(\bm 0,\bm V_\beta),
\]
where
\[
\bm V_\beta
=
\mathrm{Var}\!\left(
\sum_{g=1}^G
\mathbf 1\{\bm X_i\in g\}
\bm\Psi_g(Y_i)
\right),
\]
and $\bm\Psi_g$ is the pointwise limit of $\bm\Psi_{g,N}$, given by
$
\bm\Psi_g(y)
=
\bm\Sigma_X^{-1}\bm X_g(y - m_{g}).
$
\end{theorem}
The limiting influence function coincides with that of the
estimator that would be obtained if the group sample means
were observed and used in place of the reconstructed means
$\hat m_g$.
The proposed estimator therefore has the same asymptotic
covariance matrix as this benchmark, although its discretization
bias may remain nonnegligible at the $N^{-1/2}$ scale.
The following stronger grid condition makes this bias negligible,
yielding the same centered normal approximation.
\Needspace{8\baselineskip}
\begin{corollary}[Standard centered asymptotic normality]
\label{cor_centered_clt}
Suppose the conditions of Theorem~\ref{thm_bias_clt} hold.
Assume further that
$
\Delta(\mathcal T_{K_N})^2=o(N^{-1/2}).
$
Then
\[
\sqrt{N}
\bigl(
\hat{\bm{\beta}}-\bm{\beta}_0
\bigr)
\overset{d}{\longrightarrow}
\mathcal N(\bm 0,\bm V_\beta).
\]
In particular, for an evenly spaced grid, the above order conditions are satisfied if
$
N^{1/4}=o(K_N)
$.
\end{corollary}
\subsection{Variance Estimation and Feasible Inference}\label{subsec_variance}
To implement inference based on Corollary~\ref{cor_centered_clt},
we estimate the unknown asymptotic covariance matrix
$\bm V_\beta$ using only the disclosed summaries.
Since the groups form a partition, each observation belongs to exactly one group.
Together with $E[\bm\Psi_g(Y_i)\mid\bm X_i\in g]=\bm0$,
this implies that the cross-group covariance terms in $\bm V_\beta$ vanish.
Hence
\[
\bm V_\beta
=
\sum_{g=1}^G
\pi_g
E\left[
\bm\Psi_g(Y_i)
\bm\Psi_g(Y_i)'
\mid
\bm X_i\in g
\right]
=
\bm\Sigma_X^{-1}
\left[
\sum_{g=1}^G
\pi_g
\bm X_g\bm X_g'
\mathrm{Var}(Y_i-m_g\mid\bm X_i\in g)
\right]
\bm\Sigma_X^{-1}.
\]
Feasible estimation of $\bm V_\beta$ therefore requires
estimating the group-specific variance components.\footnote{
For overlapping groups, observations may contribute to several summaries,
generating cross-group covariance terms. Estimating these terms generally
requires joint information unavailable from the marginal summaries alone
\citep[see, e.g.,][]{farebrother79}. The role of the partition in sharp
identification is discussed in Section~\ref{sec_partial_identification}.
}
Write $\hat m_g=\sum_{k=1}^{K_N}\alpha_{k,N}\widehat Q_g(\tau_k)$,
where the reconstruction weights are
\begin{align*}
\alpha_{1,N}&=\tau_1+\frac{\tau_2-\tau_1}{2},\\
\alpha_{k,N}&=\frac{\tau_k-\tau_{k-1}}{2}
+\frac{\tau_{k+1}-\tau_k}{2},
\quad k=2,\ldots,K_N-1,\\
\alpha_{K_N,N}&=\frac{\tau_{K_N}-\tau_{K_N-1}}{2}
+(1-\tau_{K_N}).
\end{align*}
They are nonnegative and sum to one.
Because $\hat m_g$ is a weighted sum of the disclosed sample
quantiles, we estimate its variance using an asymptotic
approximation to their covariances\footnote{Supplemental Material~L provides the derivation of this variance estimator.}:
\[
\widehat{\mathrm{Var}}(\hat m_g)
=\frac{1}{n_g}\sum_{k=1}^{K_N}\sum_{l=1}^{K_N}
\alpha_{k,N}\alpha_{l,N}
\frac{\min(\tau_k,\tau_l)-\tau_k\tau_l}
{\hat f_g(\widehat Q_g(\tau_k))\,
\hat f_g(\widehat Q_g(\tau_l))},
\]
where the density estimates are constructed using the identity
$f_g(Q_g(\tau))=1/Q_g'(\tau)$:
\[
\hat f_g(\widehat Q_g(\tau_k))
=\frac{1}{\widehat{Q_g'}(\tau_k)},
\qquad
\widehat{Q_g'}(\tau_k)
=\frac{\widehat Q_g(\tau_{k+1})-\widehat Q_g(\tau_{k-1})}
{\tau_{k+1}-\tau_{k-1}}
\]
for $k=2,\ldots,K_N-1$.
At $\tau_1$ and $\tau_{K_N}$, we use forward and backward
differences, respectively.
Combining these group-level variance estimates, we estimate
the covariance matrix of $\hat{\bm\beta}$ by
\[
\widehat{\bm V}_\beta
=\left(\sum_{g=1}^G n_g\bm X_g\bm X_g'\right)^{-1}
\left(\sum_{g=1}^G n_g^2\bm X_g\bm X_g'
\widehat{\mathrm{Var}}(\hat m_g)\right)
\left(\sum_{g=1}^G n_g\bm X_g\bm X_g'\right)^{-1}.
\]
All components in $\widehat{\bm V}_\beta$ use disclosed summaries only.
Under the conditions stated below,
$n_g\widehat{\mathrm{Var}}(\hat m_g)$ consistently estimates
$\mathrm{Var}(Y_i-m_g\mid\bm X_i\in g)$, the group-specific variance
component appearing in $\bm V_\beta$.
Accordingly, $N\widehat{\bm V}_\beta$ consistently estimates
the asymptotic covariance matrix $\bm V_\beta$.
The finite-difference estimator $ \widehat{Q_g'}(\tau_k)$ divides differences of sample
quantiles by the corresponding differences in quantile indices.
Small spacings can therefore amplify sampling errors in the
derivative estimates.
To control this effect, define the minimum spacing
\[
\underline\Delta(\mathcal T_{K_N})
=
\min_{1\le k\le K_N-1}(\tau_{k+1}-\tau_k).
\]
The additional condition
$N^{-1/2}=o(\underline\Delta(\mathcal T_{K_N}))$
in the following proposition is sufficient to make this
sampling-error contribution vanish.
\begin{proposition}[Consistency of the variance estimator]
\label{prop_var}
Under the conditions of Proposition~\ref{prop_rootN} with Assumption \ref{ass_smooth}(ii), suppose that
$
N^{-1/2} = o(\underline\Delta(\mathcal T_{K_N}))
$.
Then
\[
N \widehat{\bm V}_\beta
\xrightarrow{p}
\bm V_\beta.
\]
\end{proposition}
\begin{remark}
Under an evenly spaced grid, $\tau_k = k/(K_N+1)$, we have
$
\underline\Delta(\mathcal T_{K_N})
=
\frac{1}{K_N+1}.
$
Hence, the condition
$
N^{-1/2} = o(\underline\Delta(\mathcal T_{K_N}))
$
reduces to
$
K_N = o(N^{1/2}).
$
\end{remark}
Together, Corollary~\ref{cor_centered_clt} and Proposition~\ref{prop_var}
give a feasible centered normal approximation. For an evenly spaced grid,
their conditions require $N^{1/4}=o(K_N)$ and $K_N=o(N^{1/2})$.
The finite-difference implementation requires at least two disclosed
quantiles.
The minimum-spacing condition is specific to the finite-difference
variance estimator.
Supplemental Material~H gives an alternative density-free variance
estimator based on reconstruction of the conditional second moment.
That estimator does not require the minimum-spacing condition;
its consistency is established for quantile grids becoming dense.
We retain the finite-difference procedure as the baseline in the
simulations, applications, and \texttt{qonlyreg} package because
the same approach to estimating sample-quantile covariances is
also used to implement inference under partial identification
in Section~\ref{sec_fixedDC}.
\section{Fixed Quantile Disclosure: Non-identification and Partial Identification}\label{sec_fixedDC}
The preceding asymptotic results require increasingly dense quantile disclosure.
We now study identification and inference when only a fixed finite set of quantiles is disclosed.
\subsection{Non-identification}\label{subsec_nonID}
Under fixed quantile disclosure, the regression coefficients are not generally point identified.
\begin{theorem}[Non-identification under fixed quantile disclosure]
\label{thm_nonidentification}
Suppose that $\bm\Sigma_X=\sum_{g=1}^G\pi_g\bm X_g\bm X_g'$ is positive definite, and let $\mathcal T_K=\{\tau_1,\ldots,\tau_K\}$ be any fixed finite disclosure design.
Then there exist two distinct joint distributions of $(Y,\bm X)$ with the same marginal distribution of $\bm X$ such that, for $\ell=1,2$, the group-specific conditional means satisfy $m_g^{(\ell)}=\bm X_g'\bm\beta_0^{(\ell)}$ for all $g$, and the conditional quantile functions satisfy Assumption~\ref{ass_smooth}(ii), with
\[
Q_g^{(1)}(\tau_k)=Q_g^{(2)}(\tau_k)
\quad(g=1,\ldots,G;\ k=1,\ldots,K),
\qquad
\bm\beta_0^{(1)}\ne\bm\beta_0^{(2)}.
\]
Consequently, $\bm\beta_0$ is not point identified from any fixed finite collection of disclosed quantiles.
\end{theorem}
Theorem~\ref{thm_nonidentification} isolates an identification problem caused by disclosure: even population quantiles need not determine the group means or regression coefficients.
Increasing the number of disclosed quantiles alone need not resolve it.
If $\liminf_{N\to\infty}\Delta(\mathcal T_{K_N})>0$, distinct observationally equivalent distributions with different coefficients exist for every sufficiently large $N$; these distributions may depend on $N$ (Supplemental Material~K).
\subsection{Partial Identification}\label{sec_partial_identification}
The non-identification result above motivates a partial-identification
analysis under fixed quantile disclosure.
We first derive sharp bounds for the group means and then use
these bounds to characterize the identified set for the regression
coefficients.
For partial identification, our aim is to derive bounds on group
means using only the probability restrictions implied by the
population quantiles and known support bounds.
We do not impose the smoothness or positive-density conditions
used in the asymptotic analysis on candidate distributions.
For the remainder of this section, assume that known bounds
$\underline y_g$ and $\overline y_g$ satisfy
$
P(\underline y_g\le Y\le\overline y_g
\mid\bm X\in g)=1.
$
Let $\mathcal F_g(\mathcal T_K)$ denote the class of candidate
conditional distribution functions $F$ having support contained
in $[\underline y_g,\overline y_g]$ and satisfying
\[
F(Q_g(\tau_k)-)
\le \tau_k
\le F(Q_g(\tau_k)),
\qquad k=1,\ldots,K,
\]
where $F(y-)=\lim_{z\uparrow y}F(z)$.
In defining $\mathcal F_g(\mathcal T_K)$, we hold fixed the
population values $Q_g(\tau_k)$ defined under the
generalized-inverse convention in
Section~\ref{sec_model_estimation}, but do not require them to
satisfy
$
Q_g(\tau_k)=\inf\{y:F(y)\ge\tau_k\}
$
for each candidate distribution $F$.
To characterize the means attainable under
$\mathcal F_g(\mathcal T_K)$, we introduce a class of
nondecreasing functions.
Augment the fixed grid by $\tau_0=0$ and $\tau_{K+1}=1$,
and set $Q_g(\tau_0)=\underline y_g$ and
$Q_g(\tau_{K+1})=\overline y_g$ for notational convenience.
Let $\mathcal Q_g(\mathcal T_K)$ be the set of nondecreasing
functions $Q:[0,1]\to\mathbb R$ satisfying
$Q(\tau_k)=Q_g(\tau_k)$ for $k=0,\ldots,K+1$.
No left- or right-continuity is required for these functions.
Define
\[
\mathcal M_g(\mathcal T_K)
=
\left\{
\int_0^1 Q(\tau)\,d\tau:
Q\in\mathcal Q_g(\mathcal T_K)
\right\}.
\]
The following theorem shows that $\mathcal M_g(\mathcal T_K)$
is precisely the set of means attainable by distributions in
$\mathcal F_g(\mathcal T_K)$ and gives its sharp lower and
upper bounds.
By monotonicity, any $Q\in\mathcal Q_g(\mathcal T_K)$ satisfies
$Q_g(\tau_k)\le Q(\tau)\le Q_g(\tau_{k+1})$
for $\tau\in(\tau_k,\tau_{k+1})$.
Integrating the step functions that take the left- and
right-endpoint quantile values on each interval gives the lower and upper bounds for the mean.
\begin{theorem}[Sharp bounds for the mean]
\label{thm_mean_bounds}
The set $\mathcal M_g(\mathcal T_K)$ coincides with the
identified set for the group mean $m_g$ and is given by
\[
\mathcal M_g(\mathcal T_K)
=
\left\{
\int_{\mathbb R} y\,dF(y):
F\in\mathcal F_g(\mathcal T_K)
\right\}
=
[\underline m_g,\overline m_g],
\]
where
$
\underline m_g
=
\sum_{k=0}^{K}
(\tau_{k+1}-\tau_k)Q_g(\tau_k)
$
and
$
\overline m_g
=
\sum_{k=0}^{K}
(\tau_{k+1}-\tau_k)Q_g(\tau_{k+1}).
$
These bounds are sharp: every value in
$[\underline m_g,\overline m_g]$, including both endpoints,
is attainable as the mean of a distribution in
$\mathcal F_g(\mathcal T_K)$.
\end{theorem}
The following corollary characterizes the width of this identified
interval and provides an upper bound in terms of the maximal spacing $\Delta(\mathcal T_K)$.
\begin{corollary}[Identification Width]
\label{cor_width}
The width of the identified interval satisfies
\[
\overline m_g-\underline m_g
=\sum_{k=0}^K(\tau_{k+1}-\tau_k)
\bigl(Q_g(\tau_{k+1})-Q_g(\tau_k)\bigr)
\le(\overline y_g-\underline y_g)\Delta(\mathcal T_K).
\]
\end{corollary}
For a fixed number of disclosed quantiles, evenly spaced grids
minimize the maximal-spacing bound on discretization error
(Corollary~\ref{cor_optimal_design}) and the upper bound
$(\overline y_g-\underline y_g)\Delta(\mathcal T_K)$ on
the width of the identified interval.
They need not minimize the width itself, which also depends on
the increments of $Q_g$ between adjacent quantile indices.
A nonuniform grid that places more quantiles where $Q_g$ changes
rapidly may therefore yield a narrower identified interval.
The bounds on the group means translate into restrictions on
the regression coefficients through the relation
$m_g=\bm X_g'\bm\beta_0$.
Over a parameter space $\Theta\subset\mathbb R^p$, we therefore define
\[
\mathcal B(\mathcal T_K)
=\left\{\bm\beta\in\Theta:
\underline m_g\le\bm X_g'\bm\beta\le\overline m_g
\text{ for all }g=1,\ldots,G\right\}.
\]
The following proposition shows that every
$\bm\beta\in\mathcal B(\mathcal T_K)$ can be attained by
group-specific distributions that match the disclosed quantiles
and have means $\bm X_g'\bm\beta$.
\begin{proposition}[Sharp Partial Identification of $\bm\beta_0$]
\label{prop_beta}
Suppose $\Theta$ is compact and convex.
Then $\mathcal B(\mathcal T_K)$ is the sharp identified set
for $\bm\beta_0$ and is compact and convex.
\end{proposition}
Sharpness relies on the partition structure, which permits the group-specific conditional distributions to be specified separately and combined into a joint distribution.
With overlapping groups, cross-group compatibility imposes additional restrictions, so the same group-by-group bounds need not characterize the sharp set.
\subsection{Estimation and Inference}\label{subsec_inference}
Replace $Q_g(\tau_k)$ by $\widehat Q_g(\tau_k)$ in Theorem~\ref{thm_mean_bounds}, retaining the known support bounds as the boundary-node values, to obtain $\widehat{\underline m}_g$ and $\widehat{\overline m}_g$.
The estimated identified set is
\[
\widehat{\mathcal B}(\mathcal T_K)
=\left\{\bm\beta\in\Theta:
\widehat{\underline m}_g\le\bm X_g'\bm\beta
\le\widehat{\overline m}_g
\text{ for all }g=1,\ldots,G\right\}.
\]
Proposition~\ref{prop_set_consistency} in Supplemental Material~E gives sufficient conditions for its Hausdorff consistency.
The estimated group-mean bounds inherit sampling error from the
disclosed sample quantiles.
Consequently, the plug-in set $\widehat{\mathcal B}(\mathcal T_K)$
need not contain the population identified set
$\mathcal B(\mathcal T_K)$.
We construct a confidence region by adjusting the estimated
lower bounds downward and the estimated upper bounds upward.
To calibrate these adjustments simultaneously across all groups,
we use the joint asymptotic distribution of the estimated bounds.
Specifically, let
\[
\bm s
=
(\underline m_1,\overline m_1,\ldots,
\underline m_G,\overline m_G)',
\]
and let $\widehat{\bm s}$ be its sample analog.
Under the conditions of Lemma~\ref{lem_joint_bound_normality},
\[
\sqrt N(\widehat{\bm s}-\bm s)
\xrightarrow{d}\mathcal N(\bm0,\bm\Omega).
\]
The partition structure makes $\bm\Omega$ block diagonal; its formula in terms of group shares and quantile densities is given in Supplemental Material~E.
A plug-in estimator $\widehat{\bm\Omega}$ replaces these quantities by $n_g/N$ and
$\hat f_g(\widehat Q_g(\tau_k))$.
The asymptotic coverage guarantee requires
$\widehat{\bm\Omega}\xrightarrow{p}\bm\Omega$ as $N \to \infty$.
With fixed quantile disclosure, the finite-difference density
estimates $\hat f_g(\widehat Q_g(\tau_k))$ need not consistently
estimate $f_g(Q_g(\tau_k))$, so consistency of the resulting
plug-in estimator $\widehat{\bm\Omega}$ is not generally guaranteed.
Consistency holds in special cases, such as when each $Q_g$
is linear on all intervals used for differencing.
Supplemental Material~E discusses consistency using additional
disclosed covariance or density estimates, and possible
extensions using quantiles in shrinking neighborhoods.
In the partial-identification simulations and applications,
we construct $\widehat{\bm\Omega}$ as a plug-in approximation
using finite-difference density estimates.
Exact support endpoints, when known, are used as boundary
nodes, whereas assumed support bounds do not enter the
finite-difference estimation.
Let $\widehat\omega_j^2$ be the $j$th diagonal element of $\widehat{\bm\Omega}$.
Compute the $1-\alpha$ quantile $\hat q_{1-\alpha}$ of $\max_{1\le j\le2G}|Z_j/\widehat\omega_j|$ for $\bm Z\sim\mathcal N(\bm0,\widehat{\bm\Omega})$ by Monte Carlo simulation.
Enlarging the estimated constraints gives a feasible simultaneous confidence region for $\bm\beta_0$:
\[
\widehat{\mathcal C}_{1-\alpha}
=\left\{\bm\beta\in\Theta:
\widehat{\underline m}_g
-\hat q_{1-\alpha}\frac{\widehat\omega_{2g-1}}{\sqrt N}
\le\bm X_g'\bm\beta\le
\widehat{\overline m}_g
+\hat q_{1-\alpha}\frac{\widehat\omega_{2g}}{\sqrt N}
\text{ for all }g\right\}.
\]
Under the conditions of Theorem~\ref{thm_confidence_region_coverage}, including covariance consistency,
\[
\liminf_{N\to\infty}
P\bigl(\mathcal B(\mathcal T_K)
\subseteq\widehat{\mathcal C}_{1-\alpha}\bigr)
\ge1-\alpha.
\]
Thus the region covers the entire identified set with asymptotic probability at least $1-\alpha$.
A confidence interval for the $j$th coefficient is obtained by projecting
$\widehat{\mathcal C}_{1-\alpha}$ onto the $j$th coordinate axis.
Its lower and upper endpoints are
\[
\hat\beta_j^L
=\min_{\bm\beta\in\widehat{\mathcal C}_{1-\alpha}}\beta_j,
\qquad
\hat\beta_j^U
=\max_{\bm\beta\in\widehat{\mathcal C}_{1-\alpha}}\beta_j.
\]
The lower and upper endpoints are obtained by solving convex optimization problems.
If $\Theta$ is defined by finitely many linear equality and inequality constraints, these problems are linear programs.
\section{Simulation Study}\label{sec_simulation}
We examine finite-sample estimation accuracy and confidence coverage
under increasing and fixed quantile disclosure.
All simulations and empirical analyses use the \texttt{qonlyreg}
R package developed for this paper.
\subsection{Point Estimation}
For each replication, we generate data according to the model
\[
Y_i=\beta_0+\beta_1X_i+u_i,
\qquad
u_i\mid X_i\in g\sim\mathcal N(0,\sigma_g^2),
\qquad
\sigma_g=\exp(0.5X_g),
\]
with $(\beta_0,\beta_1)=(1,2)$ and $G=10$ equally sized groups.
For each observation in group $g$, we set $X_i=X_g$, where
the group-level values $X_g$ are equally spaced over $[-1,1]$.
We consider $N\in\{2000,20000\}$ and $K\in\{3,5,7,10,15,20,50\}$.
For each of $M=10{,}000$ replications, the same sample is used across all values of $K$ and three disclosure designs: an evenly spaced grid and two progressively more right-tail-coarse grids.
The latter have larger maximal spacing $\Delta(\mathcal T_K)$ for a given $K$; their definitions appear in Supplemental Material~D.
The Gaussian design assesses robustness outside Assumption~\ref{ass_smooth}: its support is unbounded and its quantile derivatives diverge at the endpoints.
Supplemental Material~D also reports a bounded-support design satisfying the maintained conditions, additional sample sizes, and results indexed by $\Delta(\mathcal T_K)$.
Under the evenly spaced grid, symmetry makes the population mean-reconstruction error zero for every finite $K$.
Its finite-sample performance therefore reflects sample-quantile and feasible variance-estimation error, whereas the asymmetric grids generally induce population reconstruction bias.
[Figure~\ref{fig_simulation} here ]
Figure~\ref{fig_simulation} reports results for $\beta_1$.
Bias and MSE fall as $K$ increases, and grids with larger gaps perform worse for a given $K$.
Evenly spaced grids attain coverage close to 95\% with relatively few quantiles; right-tail-coarse grids require substantially more.
For the asymmetric grids, increasing $N$ alone cannot remove fixed-$K$ reconstruction bias: intervals can concentrate around a biased estimator and substantially undercover.
Supplemental Material~D provides numerical coverage results for evenly spaced grids (Table~\ref{tab:coverage_even}) and a Laplace-error experiment that violates Assumption~\ref{ass_smooth}(ii).
The grid ranking persists under Laplace errors, but more quantiles are needed for coverage to approach its nominal level.
\subsection{Partial Identification}
For fixed-disclosure inference, we retain the preceding design but replace the errors by
$u_i\mid X_i\in g\sim\mathrm{Uniform}(-\sigma_g,\sigma_g)$ to ensure bounded support.
The true group-specific support endpoints are treated as known
and are used both to construct the group-mean bounds and as
boundary nodes in finite-difference density estimation.
Because each $Q_g$ is linear under the uniform design, the
finite-difference density estimates
$\hat f_g(\widehat Q_g(\tau_k))$ used to construct
$\widehat{\bm\Omega}$ are consistent even with fixed $K$.
With evenly spaced grids and fixed $K\in\{3,9,19\}$, we examine coverage of the entire true identified set and the contraction of projection intervals.
[Table~\ref{tab:partial_identification_region_coverage_even} here ]
Table~\ref{tab:partial_identification_region_coverage_even} shows slight undercoverage when $N=1{,}000$, with coverage approaching its nominal level as $N$ increases.
Projection intervals become shorter with larger $N$, reflecting contraction toward the identified set, and with larger $K$, reflecting sharper identification.
Supplemental Material~D illustrates this geometry (Figure~\ref{fig_simulation_partial}) and reports alternative grid designs.
\section{Empirical Illustration}\label{sec_empirical}
Three applications illustrate quantile-only and mixed disclosure.
Conventional intervals account for sampling uncertainty but not finite-grid mean-reconstruction bias.
The simultaneous regions instead allow all group means consistent with the disclosed summaries and support restrictions.
\subsection{Educational Achievement}
We use publicly released NAEP summaries for the 2024 Grade~8
mathematics assessment.\footnote{All NAEP summary statistics used in
this empirical illustration were retrieved from the NAEP Data Explorer
on May 19, 2026:
\url{https://www.nationsreportcard.gov/ndecore/xplore/NDE}.}
The 50 states, District of Columbia, and Puerto Rico each contribute economically disadvantaged and non-economically disadvantaged cells, giving $G=104$ groups.
The release provides group sizes, means, and selected percentiles rather than individual-level data in this setting.
We reserve the published means for validation: point estimation uses only the disclosed quantiles, while partial identification additionally uses the known score bounds $[0,500]$.
For the group mean mathematics score $m_g$, we specify
\[
m_g=\alpha_{s_g}+\beta\,\mathrm{nonEDS}_g,
\]
where $s_g$ denotes the jurisdiction to which group $g$ belongs,
$\alpha_{s_g}$ is a jurisdiction-specific intercept, and
$\mathrm{nonEDS}_g$ indicates non-economically disadvantaged status.
Thus, $\beta$ measures the score difference after controlling for state fixed effects.
We compare median-only, quartile, and five-percentile (10th, 25th, 50th, 75th, and 90th) disclosure.
With a median alone, however, the finite-difference covariance estimators used for conventional and partial-identification inference are unavailable.
[Table~\ref{tab:naep_qonlyreg} here ]
Table~\ref{tab:naep_qonlyreg} shows that all three point estimates are close to the published-mean benchmark of 28.4.
The five-percentile conventional interval contains the benchmark, whereas the quartile interval does not.
Increasing disclosure from quartiles to five percentiles reduces the width of the projection interval by about $54\%$.
Both projected regions nevertheless remain much wider than the conventional intervals, include zero, and contain the published-mean benchmark.
\subsection{Political Control and Children in Need Assessment Duration}
English local authorities assess children's needs following referrals to social care \citep{dfe_cin_methodology}; statutory guidance calls for completion within 45 working days \citep{dfe_working_together_2026}.
The Department for Education (DfE) publishes council-year assessment counts and duration quartiles, but not mean duration.
We combine these summaries for reporting years 2017--2025 with annual council-composition records (Supplemental Material~F).
Completed assessments are the underlying units.
Let $g$ index council-year groups, with group size $n_g$ equal to the corresponding assessment count.
We compare mean assessment duration around the 50\% Labour seat-share threshold.\footnote{
This is an RD-motivated fixed-window illustration.
A conventional local-polynomial RD analysis would require a
many-group extension with a shrinking bandwidth.
}
Following \citet{alonso20}, define $R_g$ as Labour's percentage share of council seats minus 50, and let
$D_g=\mathbf 1\{R_g\geq0\}$.
For a fixed window $h$, define
$\mathcal G_h=\{g:|R_g|\leq h\}$.
Within this window, we impose
\[
m_g=\alpha_h+\delta_hD_g,
\qquad g\in\mathcal G_h,
\]
where $m_g$ is the population mean assessment duration in group $g$
and $\delta_h$ is the coefficient of interest.
The primary window is $h=4$ percentage points, with $h=3$ and $h=5$
as sensitivity specifications; estimation weights groups by
assessment counts $n_g$.
Using the known lower endpoint $Q_g(0)=0$, the disclosed quartiles,
and constant extrapolation above the 75th percentile,
reconstruction gives
\[
\hat m_g
=0.125\times0
+0.250\widehat Q_g(0.25)
+0.250\widehat Q_g(0.50)
+0.375\widehat Q_g(0.75).
\]
For partial identification, we retain the known lower support endpoint $Q_g(0)=0$ and impose the upper support bound $Y_g\leq U$, with $U\in\{150,175,200\}$ days.
These upper bounds are sensitivity assumptions, not administrative maximum durations.
We project the 95\% simultaneous region from Section~\ref{sec_fixedDC} onto $\delta_h$.
[Table~\ref{tab:qonlyreg_main} here ]
Table~\ref{tab:qonlyreg_main} reports a main-window estimate of $-0.830$ days with a conventional standard error of $0.061$.
Point estimates are negative in all three windows, but every projected interval is wide and contains zero across window and upper-support specifications.
Figure~\ref{fig_cin_main} displays the disclosed quartiles and reconstructed means.
The negative point estimate is driven primarily by the lower disclosed quartile;
Supplemental Material~F reports the quantile decomposition, reporting-year
fixed effects, alternative political-control definitions, and
leave-one-council-out results. The latter retain a negative estimate in 35 of
36 exclusions; excluding Bristol yields $0.107$ days.
[Figure~\ref{fig_cin_main} here ]
\subsection{Healthcare-Associated Infections and Medical Expenditure}
\label{subsec_hai_expenditure}
To illustrate mixed disclosure, we use Table~IV of \citet{liu22_hai}, a meta-analysis of healthcare-associated infections (HAIs) in Chinese general hospitals.
Seven study--year comparisons from five publications report expenditure in Chinese yuan (CNY) for HAI and non-HAI inpatients.
Four comparisons disclose sample means and standard deviations, including three annual comparisons for 2013--2015 from \citet{li17_hai}; three disclose quartiles.
Let $Y_g^a$ denote total medical expenditure per inpatient in comparison
$g$ and HAI status $a\in\{H,C\}$, and write $\mu_g^a=E(Y_g^a)$.
Each comparison-by-status pair $(g,a)$ constitutes a group in the general
framework. With a study--year-specific intercept $\alpha_g$, we specify
\[
\mu_g^a=\alpha_g+\beta_0D_g^a,
\qquad g=1,\ldots,7,
\qquad D_g^H=1,\quad D_g^C=0,
\]
so that $\beta_0$ is the common within-comparison mean difference, $\mu_g^H-\mu_g^C$.
We use reported means directly, reconstruct quartile-based means as in Section~\ref{sec_model_estimation}, and apply sample-size-weighted minimum distance.
Inference incorporates the reported paired-difference uncertainty for the three \citet{li17_hai} comparisons.
For the others, the unavailable covariance between HAI- and non-HAI-specific summaries is set to zero as a working assumption.
Fixed-disclosure inference imposes $Y_g^a\in[0,U]$, with $U=200{,}000$ CNY in the main specification.
This upper bound is an assumption, not an established expenditure limit.
Supplemental Material~I provides details on the elimination of
study--year-specific intercepts, covariance construction,
simultaneous inference, and support sensitivity.
[Table~\ref{tab_hai_expenditure} here ]
Table~\ref{tab_hai_expenditure} shows a positive point estimate whose conventional 95\% confidence interval excludes zero.
The estimated identified set is empty: directly reported sample mean differences impose incompatible singleton constraints under the common-effect specification.
Allowing for sampling uncertainty yields a nonempty projected interval entirely above zero, but well above the point estimate.
This discrepancy may reflect finite-grid reconstruction bias in the point estimate, as well as tension with the common-effect specification, for example due to between-study heterogeneity.
The discrepancy persists across all upper support bounds considered: the confidence region remains above the point estimate and excludes zero throughout.
The region is unchanged at \(U=200{,}000\) and \(U=500{,}000\) CNY: once support is sufficiently loose, mean-reporting comparisons bind and keep the region informative.\footnote{Supplemental Material~J gives a vitamin D--tuberculosis illustration in which mean-reporting studies similarly anchor the region, which contains the point estimate.}
\section{Conclusion}\label{sec_conclude}
What can be learned about group-mean regression coefficients depends
on the information released, particularly the number and placement
of the disclosed quantiles.
Our framework accommodates quantile-only and mixed disclosure.
Increasing disclosure permits consistent, asymptotically normal estimation under the stated conditions; fixed disclosure generally yields partial identification, for which we derive sharp sets and simultaneous confidence regions.
The number and placement of quantiles affect reconstruction bias,
the feasibility of conventional inference, and identification
uncertainty, making disclosure design consequential for regression analysis.
Extensions include shape restrictions between disclosed quantiles to sharpen identification, panel and nonlinear models, more efficient weighting, and random-effects meta-analysis.
A framework with diverging $G$ would also permit local-polynomial RD methods with a shrinking bandwidth and an increasing number of groups near the cutoff.
\textbf{Acknowledgements: }
We would like to thank Yoshiyasu Rai
and
the participants of the Summer Workshop on Economic Theory 2026
for their useful comments.
We used ChatGPT (OpenAI, GPT-5) to assist with English-language editing,
the development and debugging of R code, and the exposition and
organization of mathematical arguments.
All AI-assisted output was reviewed by the authors, who take full responsibility for the content.
\textbf{Funding: }
This work was supported by JSPS KAKENHI Grant Number 24K04826.
\textbf{Disclosure Statement: }
The authors report there are no competing interests to declare.
\textbf{Data and Code Availability:}
The empirical applications use publicly available aggregate data as
well as study-level summary statistics transcribed from published articles.
Detailed information on data sources and, where applicable, data
retrieval, construction, and licensing is provided in the
application-specific README files in the replication repository.
The replication package, including the analysis data, data-retrieval
and construction scripts where applicable, simulation and empirical
analysis code, generated outputs, and detailed README files, is
available at
\url{https://github.com/masaiwasawa/QOR_Replication}.
\bibliography{bibliography.bib}
\clearpage
\section*{Figures}
\begin{figure}[!htbp]
\centering
\includegraphics[width=\linewidth]{QOR_Replication/simulation/results_smooth_qlm/plot_main_combined.png}
\caption{\label{fig_simulation}\setstretch{2}
Absolute bias, MSE, and coverage probability of nominal 95\% confidence intervals under different quantile designs.
}
\end{figure}
\clearpage
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]
{QOR_Replication/empirical_illustration/UK_election/output/rd_quantiles_and_reconstructed_mean_main_window_4pp.png}
\caption[Disclosed quantiles and reconstructed means]{\setstretch{2}
Disclosed assessment-duration quantiles and reconstructed means
around the Labour seat-share threshold.
Notes:
Horizontal segments show assessment-count-weighted averages
of reconstructed means on each side of the threshold.
Horizontal jitter is for display only.
Observations exactly at the threshold are assigned to the treated
group and shifted slightly to the right for display.
}
\label{fig_cin_main}
\end{figure}
\clearpage
\section*{Tables}
\begin{table}[H]
\centering
\begin{threeparttable}[H]
\centering
\caption{\label{tab:partial_identification_region_coverage_even} Coverage of the simultaneous confidence region under even grids}
\centering
{
\begin{tabular}[t]{cccccccccc}
\toprule
\multicolumn{1}{c}{ } & \multicolumn{3}{c}{Coverage} & \multicolumn{3}{c}{Length of $\beta_0$ interval} & \multicolumn{3}{c}{Length of $\beta_1$ interval} \\
\cmidrule(l{3pt}r{3pt}){2-4} \cmidrule(l{3pt}r{3pt}){5-7} \cmidrule(l{3pt}r{3pt}){8-10}
$N$ & $K = 3$ & $K = 9$ & $K = 19$ & $K = 3$ & $K = 9$ & $K = 19$ & $K = 3$ & $K = 9$ & $K = 19$\\
\midrule
1000 & 0.9378 & 0.9314 & 0.9296 & 0.6990 & 0.3928 & 0.2893 & 0.8680 & 0.5198 & 0.3982\\
5000 & 0.9464 & 0.9451 & 0.9454 & 0.5970 & 0.2910 & 0.1878 & 0.7079 & 0.3641 & 0.2461\\
10000 & 0.9511 & 0.9544 & 0.9540 & 0.5719 & 0.2665 & 0.1635 & 0.6685 & 0.3261 & 0.2092\\
\bottomrule
\end{tabular}
}
\begin{tablenotes}[flushleft]
\setstretch{2}
\item Notes: Coverage reports the empirical coverage probability of the simultaneous confidence region for the true identified set. The length columns report the average lengths of the projection intervals.
\end{tablenotes}
\end{threeparttable}
\end{table}
\clearpage
\begin{table}[!h]
\centering
\begin{threeparttable}
\caption{\label{tab:naep_qonlyreg}Quantile-only regression results using NAEP data}
\centering
{
\begin{tabular}[t]{lcccccc}
\toprule
\multicolumn{1}{c}{ } & \multicolumn{3}{c}{Point} & \multicolumn{3}{c}{Partial} \\
\cmidrule(l{3pt}r{3pt}){2-4} \cmidrule(l{3pt}r{3pt}){5-7}
Disclosure & Estimate & CI & Covered & Projection CI & Width & Covered\\
\midrule
Median & 29.7 & -- & -- & -- & -- & --\\
Quartiles & 29.3 & {}[28.6, 30.1] & No & {}[-104.7, 140.0] & 244.6 & Yes\\
Five percentiles & 28.7 & {}[28.0, 29.4] & Yes & {}[-37.7, 75.1] & 112.8 & Yes\\
\bottomrule
\end{tabular}
}
\begin{tablenotes}[flushleft]
\setstretch{2}
\item Notes: Benchmark estimate is 28.4. The benchmark estimate is obtained by regressing the published group mean scores on the non-EDS indicator and state fixed effects. “Covered” indicates whether the benchmark estimate
based on the published mean scores is contained
in the corresponding interval.
\end{tablenotes}
\end{threeparttable}
\end{table}
\clearpage
\begin{table}[!h]
\centering
\begin{threeparttable}
\caption{Labour majority and assessment duration: fixed-window qonlyreg results}
\label{tab:qonlyreg_main}
{
\begin{tabular}{cccccccc}
\toprule
$h$ & $|\mathcal G_h|$ & Co. & T / C & Est. (SE) & $U=150$ & $U=175$ & $U=200$ \\
\midrule
$\pm 3$ & 77 & 32 & 43 / 34 & -1.638 (0.065) & [-23.62, 17.38] & [-29.87, 23.63] & [-36.12, 29.88] \\
$\pm 4$ & 88 & 36 & 51 / 37 & -0.830 (0.061) & [-22.12, 17.39] & [-28.37, 23.64] & [-34.62, 29.89] \\
$\pm 5$ & 129 & 42 & 74 / 55 & -2.045 (0.050) & [-22.17, 5.03] & [-28.42, 11.28] & [-34.67, 17.53] \\
\bottomrule
\end{tabular}
}
\begin{tablenotes}[flushleft]
\setstretch{2}
\item Notes: $h$ is the window half-width in percentage points; $|\mathcal G_h|$ and Co.\ are the numbers of council-year groups and councils; T/C gives treated/control group counts. Est.\ (SE) reports the Labour-majority coefficient (standard error). Brackets give projections of the 95\% simultaneous confidence region under $Q_g(0)=0$ and upper support bound $U$ (days). The main specification uses $h=4$.
\end{tablenotes}
\end{threeparttable}
\end{table}
\clearpage
\begin{table}[htbp]
\centering
\begin{threeparttable}
\caption{HAIs and medical expenditure in CNY per inpatient}
\label{tab_hai_expenditure}
{
\begin{tabular}{@{}lrr@{}}
\toprule
Method & Estimate & Interval \\
\midrule
Point estimator
& 21,250.23
& [19,435.27, 23,065.20] \\
Estimated identified set
&
& Empty set \\
Feasible confidence region
&
& [28,894.94, 38,163.10] \\
\bottomrule
\end{tabular}
}
\begin{tablenotes}[flushleft]
\setstretch{2}
\item \textit{Notes:}
Positive values indicate higher expenditure among HAI patients.
The point-estimator interval is a conventional 95\% confidence interval.
The estimated identified set and the 95 percent simultaneous confidence region are based on the support restriction
\(Y_g^a\in[0,\,200\,000]\) CNY.
\end{tablenotes}
\end{threeparttable}
\end{table}
\clearpage
\begin{center} {\Large
Supplemental material for the paper entitled \\
"Quantile-Only Regression: Inference and Partial Identification under Quantile Disclosure"
}
\end{center}
\clubpenalty=150 \widowpenalty=150 \displaywidowpenalty=50