Introduction

This paper establishes a common minimax precision frontier for estimation and honest inference at a latent regression discontinuity cutoff. Treatment assignment records which side of the latent cutoff an individual occupies, while the researcher observes a score contaminated by Gaussian measurement error. On a specified bounded-support benchmark, we characterize both worst-case expected absolute estimation error and worst-case expected length of connected confidence intervals with uniform $90\%$ coverage. The comparison holds simultaneously for every sample size $n\ge 2$, where $n$ denotes the number of independent observations, and every known noise standard deviation $\sigma \in [0,1]$, with constants independent of sample size and noise scale.

The experiment observes treatment, a binary outcome, and the contaminated score. Assignment follows the latent score, and the target is the difference between the two conditional potential-outcome means at the latent cutoff. Recorded assignment therefore separates the sample into two true-side arms, each of which presents an endpoint estimation problem. This structure connects continuity-based regression discontinuity (Hahn et al., 2001; Cattaneo et al., 2022) to statistical recovery under measurement error. In the closely related observed-treatment experiment, Pei et al. (2017) establish identification under Gaussian measurement error and local density positivity. Our question concerns the precision available from a finite sample under quantitative restrictions on the latent law.

Those restrictions define the domain of the matched frontier. The latent score has known support $[-1,1]$ and a density continuous on that interval, bounded everywhere between $1/4$ and $3/4$. Both potential outcomes are binary. Their specified continuous conditional means also lie between $1/4$ and $3/4$, and each has full-interval Hölder seminorm at most two for a known exponent $\beta \in (0,1]$, the smoothness parameter. The standardized Gaussian error is independent of the latent score and both potential outcomes. The error distribution, its scale, the support, and the smoothness exponent are public inputs. These conditions specify the benchmark in Definition 4; the uniform estimation and coverage criteria range over every law in that class.

Our first result constructs an observable estimator and a finite-sample honest interval on this benchmark. Positive polynomial weights concentrate near the cutoff within each reflected treatment arm. A finite inverse-heat transform corrects these weights for the known Gaussian contamination. Outcome-marked and unmarked sample averages then estimate the corresponding latent weighted averages, and stabilized, projected ratios recover the arm means. A public radius combines a Hölder approximation bound with a Gaussian second-moment bound. Deterministic minimization of that radius over a finite candidate set selects the estimator and interval. Theorem 1 establishes uniform coverage of at least $0.9$, bounds worst expected absolute error by the selected radius, and bounds worst expected interval length by twice that radius.

Our second result shows that this construction attains the optimal order for both decision criteria. Theorem 2 characterizes the common rate through a unique scalar resolution equation, with comparison constants depending only on the fixed Hölder exponent. Matching lower bounds concern the complete observed experiment and apply to every measurable estimator and every admissible honest connected interval, including procedures with independent randomization. The witnesses combine full-interval Hölder regularity, separation of the cutoff contrast, and polynomial cancellation under Gaussian convolution.

The resolution equation distinguishes three regimes. When the noise scale is at most $n^{-1/(2\beta +1)}$, both criteria have the direct-experiment order $$n^{-\beta /(2\beta +1)}.$$ On the intermediate branch specified in Theorem 2, their common order is $$\frac{{\sigma }^{\beta }}{[1+\log (n{\sigma }^{2\beta +1}){]}^{3\beta /2}}.$$ For each fixed positive noise scale, the compact-support branch eventually gives $${\left\{\frac{\log \log (en)}{\log (en)}\right\}}^{2\beta },$$ with comparison constants in this fixed-noise specialization permitted to depend on that noise level. The finite-sample root characterization covers both interfaces, zero noise, and every shrinking-noise sequence. For example, with Lipschitz conditional means and $\sigma =n^{-1/6}$, the intermediate branch applies for every $n\ge 2$.

The construction draws on compact-support deconvolution and Gaussian moment inversion. Meister (2007) analyzes density recovery under known compact support, while Wu et al. (2020) uses Gaussian polynomial inversion to estimate finite mixing distributions. Here polynomial inversion targets positive endpoint averages that are converted into conditional means through arm ratios. The noise-scale comparison is also related to shrinking-noise density estimation (Delaigle, 2008) and distribution estimation under Wasserstein loss (Durot et al., 2026). Our target and losses give these comparisons their specific decision-theoretic meaning. The honest interval construction follows the smoothness-based principle of combining worst-case approximation bias with stochastic uncertainty (Armstrong et al., 2018; Armstrong et al., 2020).

The benchmark has two further interpretations. First, Proposition 1 embeds it into a larger Gaussian class with local density positivity and continuous conditional means near the cutoff, preserving both the target and the randomized observed experiment. This embedding transfers the benchmark lower bounds to the larger class. Second, a prospective eligibility-effect design links administrative true-side assignment, a contaminated eligibility score, and a prespecified binary outcome. Under externally justified Gaussian calibration, a known bounded latent range, independent sampling, and the benchmark’s global density and conditional-mean restrictions, the results supply a precision benchmark for that design.

Supporting details for the main results appear in the appendix containing Theorem 5 and Proposition 2.

The paper proceeds through related work; the experiment, assumptions, and decision criteria; observable estimation and finite-sample inference; uniform minimax rates across noise scales; and interpretation and extensions. The appendices develop the polynomial construction, observed-experiment lower bounds, rate comparisons and verification scope, and proofs of the main results.

Related work

The relationship between regression discontinuity with measurement error and the deconvolution literature depends on the assignment mechanism, the target, and the criterion used to evaluate statistical procedures. Our observed-treatment experiment records the side of the latent cutoff and a Gaussian-contaminated score. The benchmark in Definition 4 combines known compact support, quantitative density bounds, binary outcomes, and a fixed Hölder bound on the potential-outcome means. Its decision problem evaluates estimators by worst-case expected absolute error and connected closed confidence intervals by worst-case expected length, subject to coverage of at least $0.9$ for every benchmark law at each sample size. Three comparisons clarify how these criteria relate to existing results.

Identification and cutoff decision criteria

The closest identification comparison is Pei et al. (2017). Their result identifies the sharp regression discontinuity effect from the observed outcome, treatment status, and mismeasured assignment variable under nondifferential measurement error, positive latent-score density near the cutoff, and Gaussian error. Observed treatment supplies information about the latent cutoff side, while Gaussian convolution permits recovery of the relevant latent distributions. Identification concerns recovery from the population observable law. The present benchmark evaluates the precision available from a finite sample under explicit quantitative restrictions. In particular, the known noise scale and the density and smoothness conditions in Definition 4 specify the class over which estimation risk and interval length are compared.

A complementary inference comparison comes from Dong (2018). For observed treatment and a known error density, Theorem 6 establishes asymptotic normality of the deconvolution local-linear cutoff estimator after centering by its bias and standardizing by its variance. For Gaussian error, the prescribed bandwidth has order $(\log n{)}^{-1/2}$. The theorem also imposes kernel and transform-integrability conditions and a Lyapunov condition; Assumption 11 includes twice differentiable latent density and one-sided regression smoothness near the cutoff. These conditions support a pointwise distributional approximation for local-linear inference. Our criteria instead place the supremum over the specified benchmark class inside the risk and coverage requirements. The finite-sample guarantee in Theorem 1 and the rate comparison in Theorem 2 address these uniform criteria under the benchmark’s density bounds and Hölder restrictions. This theorem-level comparison relies on the cited-source attestation; its provenance limitation is recorded in the verification appendix, and it has no proof-use in our results.

Assignment and target distinctions also matter within the regression discontinuity literature. Eckles et al. (2025) exploit known exogenous noise in the score used to assign treatment. Their approach uses the resulting assignment randomness to estimate weighted treatment effects associated with changes in the assignment rule. In our experiment, assignment follows the latent score, and measurement error enters the score available to the researcher. The target is the potential-outcome contrast at the latent cutoff. Auxiliary information provides another route: Davezies et al. (2017) study a two-sided fuzzy design with nondifferential measurement error and an auxiliary sample of treated individuals containing both true and noisy scores. Their target is a local average treatment effect, and their auxiliary observations identify features of the measurement process.

Observed-score interpretations supply a further comparison. Under correct classification of assignment and continuity of the potential-outcome means conditional on the observed score, Dong et al. (2023) identify the treatment effect for units at the observed-score cutoff. Their additional condition on treatment-effect dependence on measurement error gives a weighted-average interpretation over latent scores. These conditions align conventional regression discontinuity analysis with an observed-score target. The latent-cutoff benchmark specifies a different conditioning event and uses the recorded treatment side together with the proxy distribution to recover its endpoint contrast.

Compact support, boundary density estimation, and moments

Compact-support deconvolution provides the second decision-problem comparison. Meister (2007) obtain uniform mean integrated squared error of order $(\log n/\log \log n{)}^{-2s}$ for density estimation over a compactly supported Fourier–Sobolev class of smoothness $s$. Their procedure uses a known bound on the support and knowledge of the error characteristic function on a neighborhood of zero, where it is bounded away from zero. Thus, compact support supplies information that can improve density recovery even under partial knowledge of the error distribution. Our benchmark uses known support and a fully specified Gaussian measurement process, with Hölder smoothness imposed on conditional outcome means. The loss concerns an endpoint causal contrast, while the score density enters as a nuisance subject to continuity and quantitative bounds. These rates characterize different decision problems: integrated density recovery under Fourier–Sobolev restrictions and endpoint contrast recovery under conditional-mean Hölder restrictions.

Boundary deconvolution and Gaussian moment methods clarify the constructional relationship. Zhang et al. (2009) study a density supported on a half-line under known independent additive error. They construct boundary deconvolution kernels and analyze pointwise mean squared error for density and density-derivative estimation under ordinary-smooth and supersmooth errors; their construction can also be adapted to compact support. This is a direct endpoint antecedent, but its target is the latent density rather than a marked-to-unmarked regression ratio, and its criterion does not include honest confidence-interval length. Wu et al. (2020) estimate finite Gaussian mixing distributions under Wasserstein loss, using bounded component locations and a specified number of components; their theorem distinguishes known and unknown common variance. For known variance, Hermite polynomials supply unbiased latent-moment estimates, which their procedure projects onto a feasible moment space before reconstructing the mixing distribution. The finite inverse-heat transform in Definition 8 uses the related polynomial inversion principle to estimate endpoint-weighted averages. The arm ratios in Algorithm 1 then convert marked and unmarked averages into conditional-mean estimates. This connects Gaussian moment inversion to the cutoff target through an explicitly defined observable procedure.

Deconvolution regression gives a closer antecedent for the ratio structure. Fan et al. (1993) observe a response together with an error-contaminated covariate and estimate the latent regression function by normalizing a deconvolved response numerator by a deconvolved density denominator. They assume a known error law with nonvanishing characteristic function, a latent density bounded away from zero on a compact interval, and smoothness of both the density and regression function. For a supersmooth error of order ${\beta }_e$, their uniform squared-risk upper rate is $O((\log n{)}^{-2k/{\beta }_e})$; Gaussian error has ${\beta }_e=2$, yielding the displayed convergence rate $(\log n{)}^{-k/2}$ in root mean square, with matching minimax results under their stated classes. That result concerns regression at points in a fixed interval with a fixed error law. The present target combines two recorded treatment arms at the latent support endpoint and evaluates expected absolute error and honest interval length uniformly over the full range of $\sigma$.

Convolution-model linear functionals provide a complementary decision-theoretic comparison. Butucea et al. (2009) estimate $⟨\psi ,g⟩$ from noisy observations under a known convolution density whose Fourier transform is nowhere zero. Their adaptive oracle bound balances the squared Fourier-tail bias against a cutoff-dependent penalty, and their pointwise-density specialization recovers the usual pointwise deconvolution regimes. Pensky (2017) give upper and lower quadratic-risk bounds for general linear functionals over Sobolev signal classes with ordinary- or supersmooth Fourier behavior; the bounds match up to constants in the cases identified there and otherwise may differ by a logarithmic factor. Pointwise density is one of their applications. These results locate the inverse-problem difficulty for a single linear functional. Our endpoint estimand is instead assembled from four such marked and unmarked quantities through stabilized ratios, and the benchmark asks simultaneously for all-noise-scale absolute-risk and honest-length guarantees.

Vanishing noise and the estimation target

Shrinking-noise density estimation supplies the third comparison. Delaigle (2008) analyze mean integrated squared error for deconvolution density estimators as sample size increases and measurement-error scale decreases. Under their density-smoothness, kernel, and error assumptions, sufficiently small noise permits the corresponding error-free density rate; larger noise introduces a scale-dependent cost. This makes the comparison between noise scale and statistical resolution central to density estimation. For the present endpoint criterion, Definition 11 and Theorem 2 give the corresponding rate characterization across the benchmark noise scales, with constants uniform in sample size and noise scale at fixed benchmark smoothness.

A recent minimax comparison is Durot et al. (2026). Their deconvolution analysis estimates a signal distribution under Wasserstein-one loss over a moment-bounded class, with known error distribution and known noise scale. Under Gaussian noise with standard deviation ${\sigma }_n$, Theorem 2.4 gives the lower bound order ${\sigma }_n(\log n{)}^{-1/2}+n^{-1/2}$. Their upper bounds match when ${\sigma }_n\ge n^{-1/2+\eta }$, for fixed $\eta >0$, and when ${\sigma }_n\le n^{-1/2}$; the intervening regime carries separately stated logarithmic bounds under additional reciprocal-characteristic-function regularity. Our comparison concerns absolute error in a latent-cutoff contrast and expected length under uniform finite-sample coverage, so these distribution-recovery rates provide a regime comparison rather than a numerical benchmark for our loss.

Honest inference under smoothness restrictions

The interval criterion belongs to the optimal-recovery and smoothness-based regression literature. Donoho (1994) connect statistical estimation over restricted function classes with optimal recovery. Armstrong et al. (2018) derive finite-sample optimal confidence procedures for linear functionals in regression models with normal errors and known variance, together with uniform asymptotic extensions. Armstrong et al. (2020) construct honest intervals by incorporating worst-case smoothing bias into the critical value and bandwidth choice. Our finite-sample certificate similarly combines a public bias bound with a stochastic-error bound, here for inverse-heat averages and stabilized arm ratios. Coverage is imposed through Definition 5, and precision is evaluated by worst expected length over that same class. These choices give the endpoint estimation and inference results a common decision-theoretic criterion.

Experiment, assumptions, and decision criteria

Throughout, $|\cdot |$ denotes absolute value and $\Vert \cdot \Vert$ denotes the Euclidean norm. Generic positive constants, denoted by $C$, may depend on the fixed smoothness parameter unless another dependence is stated; their values may change between displays. We write $a\lesssim b$ when $a\le Cb$, and $a\asymp b$ when both inequalities hold. The public smoothness parameter $\beta$ lies in $(0,1]$, the known noise scale $\sigma$ lies in $[0,1]$, and the sample size $n$ is an integer at least two. Treatment indices $d$ range over $\{0,1\}$, and sample indices $i$ range from $1$ to $n$.

Latent and observed experiments

The experiment combines a latent running variable, binary potential outcomes, and an additive measurement error. Treatment follows the latent cutoff side, and the realized assignment is recorded alongside the outcome and the contaminated score. We first specify the latent law and the versions of its density and conditional means.

Under the ambient latent law $P$, the running variable $X$ has known support $[-1,1]$, the pair $(Y^0,Y^1)$ contains the untreated and treated potential outcomes, and $\epsilon$ is the standardized measurement error. The density $f(P,\cdot )$ and conditional-mean versions ${\mu }_d(P,\cdot )$ are part of this specification. Their continuous versions give a precise meaning to evaluation at the cutoff and at the support endpoints.

The observation map retains the true-side assignment $D$, selects the observed outcome $Y$ from the potential outcomes, and reports the proxy score $W$: $$D=1\{X\ge 0\},Y=(1-D)Y^0+D{Y}^1,W=X+\sigma \epsilon .$$ We consistently order the observed triple as $$O=(W,D,Y),P_{\mathrm{obs}}=P\circ O^{-1}.$$ The recorded assignment identifies the treatment arm of each observation, while the proxy supplies information about its latent distance from the cutoff. The causal target $\theta (P)$ is $$\theta (P)={\mu }_1(P,0)-{\mu }_0(P,0).$$ Continuity connects this contrast to the arm-specific limits at zero. Statistical procedures use an independent observed sample $S_n$ and may randomize through an independent uniform seed $U$. Their joint law is $${\mathbb{P}}_{P,n}=P_{\mathrm{obs}}^{\otimes n}\otimes \mathrm{Unif}[0,1],S_n=(O_1,\dots ,O_n),U\sim \mathrm{Unif}[0,1].$$ The seed permits randomized estimators and interval procedures within a common decision class. Deterministic procedures are included by taking their outputs to be constant in the seed.

Law restrictions and benchmark

The benchmark imposes five restrictions on the latent experiment: two determine the measurement-error structure, two give uniform density and mean bounds, and one controls the smoothness of the potential-outcome means. The Gaussian distribution and joint-independence conditions were stated with the observation map above. We now fix the ambient latent law and its continuous conditional-mean versions before imposing the quantitative density, mean, and Hölder restrictions.

The Gaussian classical measurement-error condition supplies a calibrated error distribution, with the public scale determining the magnitude of contamination (Pei et al., 2017). Independence specifies how this error is coupled to the latent score and potential outcomes.

The joint-independence condition in Assumption 2 is specific to this analysis. It makes the same calibrated error distribution apply across latent scores and potential-outcome realizations, so the observation map has a common additive contamination mechanism. We next fix uniform bounds on the density throughout the known support.

The density bounds are specific to this analysis: they maintain positive latent-score density across the entire support and control its maximum. Continuity is imposed on the latent interval through Definition 2; together, these requirements specify both the domain and the amount of score mass available near either side of the cutoff. A separate restriction keeps the binary conditional means uniformly interior.

These mean bounds are also specific to this analysis. They keep each conditional success probability away from zero and one throughout the latent interval and imply $\theta (P)\in [-1/2,1/2]$. Smoothness is specified separately through the full-interval Hölder seminorm $[{\mu }_d(P,\cdot ){]}_{\beta }$, which measures variation between every pair of latent-score values.

Using this seminorm, the smoothness restriction fixes a common radius for both potential-outcome mean functions.

The standard Hölder-ball restriction gives a fractional-order extension of the Lipschitz smoothness condition used in honest inference (Armstrong et al., 2020). Here the maintained radius is two, and the restriction applies on the entire latent interval, including pairs of points on opposite sides of the cutoff. Thus the density is continuous and globally bounded on its support, while the potential-outcome means satisfy an explicit full-domain Hölder bound. These conditions define the exact benchmark class.

The benchmark class ${\mathcal{M}}_{\beta }(\sigma )$ treats the Gaussian calibration, noise scale, latent support, and smoothness exponent as known inputs. Its density and conditional means vary across admissible laws. Fixing these public inputs makes the statistical comparison uniform over the latent laws satisfying the displayed restrictions, with the smoothness radius and numerical bounds held fixed.

Estimation and honest inference

For estimation, the decision class ${\mathcal{T}}_n$ consists of all measurable real-valued procedures on the randomized observed experiment. Its minimax absolute risk is $$R_{\beta }(n,\sigma )=\underset{\hat{\theta }\in {\mathcal{T}}_n}{\inf }\underset{P\in {\mathcal{M}}_{\beta }(\sigma )}{\sup }{\mathbb{E}}_{P,n}\left[|\hat{\theta }(S_n,U)-\theta (P)|\right].$$ The infimum permits every measurable estimator, including procedures chosen using the public inputs. The supremum requires a single chosen procedure to control risk across the entire benchmark class. For inference, the corresponding requirement is uniform coverage. The honest connected-interval class ${\mathcal{I}}_{\beta ,n,\sigma }$ imposes this requirement at each sample size.

Honesty requires coverage of at least $0.9$ for every law in the benchmark, with probability taken over both sampling and procedure randomization. Precision is measured by $${\mathcal{L}}_{\beta }(n,\sigma )=\underset{I\in {\mathcal{I}}_{\beta ,n,\sigma }}{\inf }\underset{P\in {\mathcal{M}}_{\beta }(\sigma )}{\sup }{\mathbb{E}}_{P,n}\left[\mathrm{length}(I(S_n,U))\right].$$ The interval $[-1,1]$ belongs to the honest class because it contains every admissible cutoff contrast. The length criterion therefore compares the precision of uniformly valid connected intervals over a nonempty decision class. Together, the minimax absolute risk $R_{\beta }(n,\sigma )$ and minimax worst expected length ${\mathcal{L}}_{\beta }(n,\sigma )$ place estimation and honest inference on the same latent-law benchmark, with common public calibration, support, and smoothness inputs.

Observable estimation and finite-sample inference

For the benchmark class in Definition 4, the public polynomial procedure combines positive averaging near the latent cutoff with an observable correction for Gaussian measurement error. The recorded assignment identifies the latent cutoff side, allowing the two arms to be treated separately. A stabilized ratio then estimates each arm’s endpoint mean, and deterministic comparisons of public radii select the effect estimate and confidence interval.

The construction begins with a positive integer $L$, the polynomial index. We use the Chebyshev polynomial $T_L$ with the following normalization.

From this family, we construct the normalized endpoint weight $K_L$. Its nonnegative values on the unit interval support latent averaging, while its polynomial extension permits evaluation at every observed proxy value.

The normalization makes $K_L$ integrate to one on $[0,1]$. The polynomial inequalities governing its concentration near zero are developed in Lemma 1. To specify the observable correction, first record the polynomial degree $m_L$ and derivative convention.

Gaussian averaging acts on a polynomial through finitely many even derivatives. The inverse-heat polynomial $Q_{L,\sigma }$ reverses that averaging using the known measurement-error scale.

Under Assumptions 1 and 2, averaging this correction over the measurement error recovers the corresponding latent polynomial weight. At zero noise, the correction equals $K_L$. For positive noise, its observable values may have either sign even though the recovered latent weight is nonnegative. The ratio construction therefore stabilizes the empirical denominator explicitly.

Both arms use the same endpoint weight after reflection onto the positive latent interval. The arm sign $s_d$ fixes this convention.

The population marked average $A_{d,L}(P)$ and unmarked average $B_{d,L}(P)$ describe the endpoint averaging that the observable correction targets. We define them before introducing their sample counterparts.

$$A_{d,L}(P)={\int }_0^1{K}_L(x){\mu }_d(P,s_{d}x)f(P,s_{d}x)dx,B_{d,L}(P)={\int }_0^1{K}_L(x)f(P,s_{d}x)dx.$$

The ratio $A_{d,L}(P)/B_{d,L}(P)$ averages the arm’s conditional mean with a positive, density-adjusted endpoint weight. Gaussian correction makes these population quantities accessible through averages of the observed triple. The outcome-marked sample average ${\hat{A}}_{d,L}$ estimates the numerator, and the unmarked sample average ${\hat{B}}_{d,L}$ estimates its normalizing mass. Their stabilized ratio gives the projected arm estimate ${\hat{\mu }}_{d,L}$. The algorithm lists the coordinates in the computational order $(D_i,Y_i,W_i)$, but their law is the consistently ordered observed triple $O_i=(W_i,D_i,Y_i)$ defined above.

The two averages retain normalization by the full sample size, so their ratio adjusts for the arm’s weighted population mass. Their expectations are $A_{d,L}(P)$ and $B_{d,L}(P)$, respectively. The denominator floor makes the estimate well defined for every observed sample, including samples with small or negative corrected mass. Projection enforces the conditional-mean range in Assumption 4.

To assemble the effect estimate and interval, we use two public quantities: the endpoint bias moment $M_{L,\beta }$ and the derivative variance cost $V_L(\sigma )$. The former measures the displacement of the latent weight from the cutoff under the maintained Hölder exponent; the latter governs the sampling term in the certificate. The candidate effect estimate ${\hat{\theta }}_L$, public radius ${\rho }_L$, and connected interval $I_L$ are defined together, including their constant fallback values.

The fallback interval contains the full target range implied by Assumption 4. Positive-index candidates combine the two projected arm estimates and add a radius accounting for endpoint approximation and sampling variability. Intersecting with $[-1,1]$ preserves coverage of the admissible target and gives a connected closed interval. Polynomial derivative bounds and the Gaussian covariance calculation are presented in Lemmas 2 and 3.

Selection takes place over a finite candidate set. Its public upper limit $L_{\max }$ is specified before observing the sample.

$$L_{\max }=\lceil n^{1/(4\beta +2)}\rceil .$$

The selected index $L_{\star }$ minimizes the public radius over this set, with the smallest index resolving ties. The selected certificate $E_{\beta }(n,\sigma )$ is that minimum radius.

The displayed public cap and selection rule define the procedures used in Theorem 2. For fixed public inputs, the selected index is fixed across samples. Consequently, its coverage guarantee follows from the corresponding fixed-index guarantee, or from the full target-range fallback when index zero is selected. Including that fallback also ensures $E_{\beta }(n,\sigma )\le 1/2$. The complete procedure has the following finite-sample certificate.

The certificate separates endpoint approximation from sampling variability. Positive latent averaging and the density lower bound give each population denominator mass of at least $1/4$, placing the empirical floor below that mass. Smoothness controls approximation through the endpoint moment, while the derivative variance cost accounts for the observable Gaussian correction. Together, these terms yield both an absolute-error bound and coverage of at least $0.9$ at each positive index.

Public selection converts these candidate guarantees into a single uniformly honest procedure over the benchmark class. Its deterministic radius bounds worst expected absolute error, and twice that radius bounds worst expected interval length. Under the stated calibration, support, density, and conditional-mean restrictions, Theorem 1 therefore supplies a finite-sample comparison of candidate precision together with a coverage guarantee for the selected interval.

Uniform minimax rates across noise scales

The public certificate in Theorem 1 bounds both estimation error and honest interval length. We now characterize the optimal scale of these two criteria across sample sizes and calibrated noise levels. The characterization uses a scalar resolution equation that balances endpoint approximation against the cost of Gaussian inversion. Its definition also identifies the observable procedures that attain the resulting rate.

The noise cost $\Psi (h,\sigma )$ summarizes Gaussian inversion cost for the rate comparison. The selected resolution $h_{\star }(n,\sigma )$ translates this cost into an endpoint approximation scale, and the rate $r_{\beta }(n,\sigma )$ is its Hölder power. The selected estimator ${\tilde{\theta }}_{\beta ,n,\sigma }$ and selected interval ${\tilde{I}}_{\beta ,n,\sigma }$ are the procedures chosen by Algorithm 2. Thus the scalar equation describes the precision achieved by the finite public selection rule.

The two noisy branches come from distinct parts of the same endpoint mechanism. In the intermediate range, the noise cost grows as $(\sigma /h{)}^{2/3}-1$, linking finer endpoint resolution to a larger Gaussian inversion cost. Below the compact-support interface $h={\sigma }^4$, the cost becomes $h^{-1/2}\{1+\log ({\sigma }^2/\sqrt{h})\}-1$. These expressions agree at the interface. The balance equation therefore connects endpoint approximation to inversion cost continuously across the two noisy regimes, using the same resolution to compare both decision criteria.

On the benchmark class of Definition 4, the following result characterizes minimax absolute risk and minimax honest expected length simultaneously. Its comparison constants are uniform in both sample size and noise scale.

Theorem 2 contains the exact branch conditions, both interfaces, and the relevant shrinking- and fixed-noise specializations. Its scalar balance connects endpoint approximation with Gaussian inversion cost, while the matched upper and lower bounds show that this one resolution governs both absolute risk and honest expected length. In the direct regime, the inversion cost vanishes at the direct resolution. In the intermediate regime, resolving the endpoint more finely than the noise scale incurs the fractional-power cost in Definition 11. The compact-support regime uses the finer-resolution cost below ${\sigma }^4$, yielding the logarithmic comparison in Theorem 2.

For reference, Table 1 collects the exact branch conditions and resolutions. The auxiliary scalars $z$ and $\tau$ are the unique solutions specified in Theorem 2.

At the direct–noisy interface, $\sigma =h_0$ and $h_{\star }=\sigma =h_0$. At the intermediate–compact interface, $\log (n{\sigma }^{4\nu })={\sigma }^{-2}-1$ within the noisy region, and $h_{\star }={\sigma }^4$; both noisy scalar formulas agree there. The common rate is $h_{\star }^{\beta }$, and its comparison with both minimax criteria uses constants depending only on $\beta$, uniformly in $n$ and $\sigma$.

The uniform characterization also distinguishes asymptotic paths. For polynomially shrinking noise $\sigma =n^{-a}$, a fixed exponent $0<a<1/(2\beta +1)$ gives the order $n^{-a\beta }(\log n{)}^{-3\beta /2}$, whereas $a\ge 1/(2\beta +1)$ places every sample size in the exact direct branch. With fixed positive noise, the rate has order $\{\log \log (en)/\log (en){\}}^{2\beta }$. The Lipschitz example $\beta =1$, $\sigma =n^{-1/6}$ sharpens this comparison: the intermediate condition holds for every $n\ge 2$, so its displayed formula describes the entire sequence. These are comparisons of rates; the exact resolution is determined by the scalar equation, and the minimax inequalities retain their comparison constants. Constants for the uniform branch comparisons depend only on $\beta$, while those for a specified asymptotic path may also depend on its fixed exponent or fixed noise level as stated in Theorem 2.

At zero noise, endpoint approximation and sampling variability balance at $h_0=n^{-1/(2\beta +1)}$, giving the Hölder power $n^{-\beta /(2\beta +1)}$. The direct-experiment specialization in Theorem 5, presented in the verification appendix, establishes this order for both criteria and its attainment by the selected estimator and honest interval. Together with the exact direct branch in Theorem 2, it shows how sufficiently precise score measurement recovers the direct-experiment rate throughout the region $\sigma \le h_0$.

Interpretation and extensions

The rate characterization in Theorem 2 describes the information available when assignment records the latent cutoff side, the latent score has known bounded support, and the Gaussian measurement-error distribution is calibrated. Recorded assignment separates the observations into the two latent-score arms. Within each arm, the cutoff is an endpoint, and the contaminated score supplies information about distance from that endpoint. These features give the endpoint weights in Definition 7 their statistical interpretation.

Known Gaussian calibration supplies the inverse-heat transform in Definition 8. The finite-sample guarantees in Theorem 1 combine this correction with the global density and conditional-mean restrictions defining Definition 4. The density bounds control the mass entering the arm ratios, while the full-interval Hölder restrictions control their approximation error. Together, these conditions support observable estimation and honest inference uniformly over the benchmark, with matching minimax orders across the stated noise scales.

Converse transfer to a larger Gaussian class

The benchmark also supplies lower bounds for a larger class specified through local density and conditional-mean conditions. To express this implication precisely, the following definition fixes the larger class, its target, and its decision criteria immediately before the embedding result.

The intuition behind Proposition 1 is inclusion of statistical decision problems with exactly the same target and data distribution on the embedded benchmark. Any estimator evaluated over the larger class faces every benchmark law. Likewise, an interval honest over the larger class satisfies the benchmark coverage requirement. Preserving the independent procedure seed extends both comparisons to randomized procedures. The benchmark lower bounds therefore remain necessary precision bounds for the larger class.

The local positivity and continuity requirements in the larger Gaussian class of Proposition 1 give the cutoff contrast a well-defined meaning, while joint independence specifies a common Gaussian contamination mechanism for the latent score and observed outcome. For positive noise scales, that proposition connects these restrictions to Assumptions 1Y, 2C, and 7 of Pei et al. (2017). The resulting interpretation is an embedding-based converse for the larger Gaussian class. Matching attainment and finite-sample honesty have the exact benchmark in Definition 4 as their domain, as established by Theorems 1 and 2.

A prospective eligibility-effect benchmark

Following the Medicaid/CHIP setting in Dong (2018), the proposed eligibility application would be organized around an eligibility cutoff.

The proposed design would link administrative eligibility assignment to a contaminated score and a prespecified binary outcome. The score would be oriented so that eligibility corresponds to its nonnegative latent side. Potential outcomes would be indexed by eligibility assignment, giving the cutoff contrast the interpretation of an eligibility effect on the chosen outcome.

Applying the benchmark would require an administrative record that identifies the latent cutoff side, together with externally justified Gaussian calibration of the score error and its known scale. The proposed error mechanism would satisfy joint independence from the latent score and both potential outcomes. A prespecified population would have a known bounded latent range, normalized to $[-1,1]$ with the cutoff at zero, and observations would follow the independent sampling experiment in Definition 4.

The remaining proposed design conditions are substantive global restrictions: a continuous latent density between $1/4$ and $3/4$ throughout the normalized interval, continuous binary potential-outcome means between the same bounds, and full-interval Hölder seminorms at most two for a specified exponent in $(0,1]$. These are the restrictions in Assumptions 3, 4, and 5. Under these maintained conditions, Theorem 1 supplies an observable interval with uniform finite-sample coverage, and Theorem 2 characterizes its attainable precision. The application thus provides a prospective design benchmark conditional on administrative linkage, calibration, support, and global regularity.

Limitations and future work

The analysis treats the Gaussian error distribution and its scale as known. Estimated calibration would introduce uncertainty into both the inverse-heat weights and the public interval radius. Extending the coverage guarantee to that setting requires an experiment that includes the calibration data and conditions controlling its estimation error.

Smoothness is also a public input. Adaptation to an unknown exponent would require procedures and coverage arguments that account for selection across smoothness classes. Empirical assessment of benchmark membership presents a separate challenge: the support, density bounds, and global potential-outcome mean restrictions concern latent quantities. Administrative and validation data could inform their plausibility, but the prospective eligibility design remains conditional on these maintained restrictions.

Additional outcome models would broaden the application beyond binary responses. Such extensions require suitable outcome-tail or moment conditions and corresponding changes to the finite-sample second-moment bounds. The present results establish estimation and honest inference for the binary-outcome benchmark.

Finally, the focused comparison in Section 2 separates three antecedents: boundary kernels estimate a truncated latent density, deconvolution regression uses a numerator–denominator ratio at fixed error law, and convolution linear-functional theory gives quadratic-risk bounds for individual functionals. None of these cited results supplies the joint criterion used here: a recorded-arm endpoint contrast with absolute-risk and honest-length guarantees uniform over the benchmark noise scales. The comparison with Zhang et al. (2009) is qualitative, covering the boundary-density target, error classes, and pointwise mean squared error criterion described in the available primary-source material. The operational size of the explicit finite-sample certificate at applied sample sizes is likewise left for a separate calibrated numerical study; the present paper establishes its validity and optimal order without reporting a design-specific grid.

Appendices

Polynomial construction and the finite-sample certificate

The achievability argument for Theorem 1 combines endpoint concentration, control of polynomial derivatives, and exact Gaussian second moments. These ingredients connect the observable correction in Definition 8 to the stabilized ratios and public intervals in Algorithm 1 and Definition 10. We present the auxiliary results in that mathematical order.

Hermite polynomials and Gaussian averaging

The Gaussian identities use the probabilists’ Hermite polynomials. We fix their normalization before stating the orthogonality result.

This normalization determines the factorial weights in Gaussian polynomial second moments. The relevant classical identity is the following.

Orthogonality separates the contributions of distinct polynomial orders under Gaussian averaging. Consequently, a finite Hermite expansion has a second moment expressed as a sum of squared coefficients with factorial weights. This structure supplies the exact second-moment identity in Lemma 3.

Positive endpoint concentration

The latent averaging step uses the endpoint polynomial weight in Definition 7. Its positivity and concentration determine how accurately a weighted arm mean approximates the corresponding cutoff mean. The following bounds quantify that concentration and its squared-integral cost.