Introduction

This paper characterizes how measurement precision and local treatment support jointly determine the accuracy of pointwise causal estimation under calibrated public density envelopes. We study the population mean response to setting a latent continuous treatment to the prespecified dose $1/2$. Treatment is observed with independent Gaussian measurement error of known standard deviation $\sigma$, the error scale. Within the bounded structural law class in Definition 3, assuming $c_{\mathrm{lo}}\le (\kappa +1)2^{\kappa }\le c_{\mathrm{hi}}$, we establish matching orders for minimax expected absolute error and minimum worst-law expected length of honest connected $(1-\alpha )$-coverage intervals for $\alpha \in (0,1/2)$. This calibrated-class characterization is uniform over the maintained range $\sigma \in [0,1/4]$, including error-free observation, for fixed public class constants and exponents in their declared ranges. An observable estimator and interval attain these orders, and structural alternatives evaluated through their induced observed distributions supply the converse under the same envelope calibration.

The model combines causal adjustment with explicit restrictions on dose availability and response regularity. Observations consist of a stratum indicator taking two values, a contaminated treatment measurement, and a realized outcome. The latent dose lies in $[0,1]$. Public constants $K=(p_{\min },c_{\mathrm{lo}},c_{\mathrm{hi}})$ use $p_{\min }$ as the minimum stratum probability and $c_{\mathrm{lo}},c_{\mathrm{hi}}$ as the density-envelope constants. Thus stratum probabilities lie in $[p_{\min },1-p_{\min }]$, while the density of the centered latent-dose displacement $t=A-1/2$ lies between $c_{\mathrm{lo}}|t{|}^{\kappa }$ and $c_{\mathrm{hi}}|t{|}^{\kappa }$, where $0<p_{\min }\le 1/2$ and $0<c_{\mathrm{lo}}\le c_{\mathrm{hi}}$. Conditional potential-outcome means satisfy radius-one Hölder increment bounds with exponent $\beta \in (0,1]$, the mean-smoothness exponent, and take values in $[0,1]$; potential outcomes are bounded in the same interval. We maintain $\kappa \in [0,2]$. Conditional exchangeability applies to the entire measurable potential-outcome schedule, consistency links its random-dose evaluation to the realized outcome, and the Gaussian error is independent of the structural coordinates. Under these conditions, Proposition 1 identifies the population causal mean even when the dose-density envelopes vanish at the evaluation point.

The main result describes three ranges of measurement precision under the envelope calibration $c_{\mathrm{lo}}\le (\kappa +1)2^{\kappa }\le c_{\mathrm{hi}}$. Let $n$ denote sample size, let $d=2\beta +\kappa +1$ be the effective smoothness-and-design exponent, and let $L_n=\log (en)$ be the logarithmic sample-size scale. The localization scale $b_{n,\sigma }$ is $$b_{n,\sigma }=\{\begin{array}{ll}{n}^{-1/d},& 0\le \sigma \le n^{-1/d},\\ \frac{\sigma }{\sqrt{\log (e+n{\sigma }^d)}},& n^{-1/d}<\sigma \le L_n^{-1/2},\\ \frac{\log (e+{\sigma }^2{L}_n)}{L_n},& L_n^{-1/2}<\sigma \le 1/4.\end{array}$$ For this calibrated class and all sufficiently large samples, Theorem 1 bounds both optimal criteria above and below by constant multiples of $b_{n,\sigma }^{\beta }$, uniformly over the known error scale and the declared public potential-outcome spaces. The direct branch balances mean smoothness against the scarcity of nearby latent doses. Gaussian Fourier inversion governs the intermediate branch, while known compact support supports polynomial localization in the third branch. For every fixed multiplicative neighborhood of either threshold, the adjacent formulas are comparable once sample size is sufficiently large. Within the calibrated class, at zero error, both criteria have order $n^{-\beta /(2\beta +\kappa +1)}$; at every fixed positive error scale in the stated range, both eventually have order $(\log \log n/\log n{)}^{\beta }$.

The upper bound comes from positive latent localization weights paired with observable inverses of the Gaussian channel. Weighted outcome and treatment moments estimate a localized mean within each stratum, and empirical stratum frequencies aggregate those estimates into the population target. A finite dictionary contains Fourier weights, polynomial inverse-heat weights, and a constant entry. The estimator selects a dictionary entry by minimizing a public expected-error certificate determined by sample size, the known error scale, the public class constant $K$, and the smoothness and design-decay exponents $\beta ,\kappa$. Thus the construction in Algorithm 1 uses one selection rule across the three regimes. Theorem 2 establishes its uniform large-sample risk bound. Dividing the same certificate by $\alpha$ calibrates a connected interval with coverage at least $1-\alpha$ for every sample size and every law in the model class, while its expected-length bound attains the frontier for sufficiently large samples, as stated in Theorem 3.

Under the envelope calibration $c_{\mathrm{lo}}\le (\kappa +1)2^{\kappa }\le c_{\mathrm{hi}}$, the matching lower bound uses Bernoulli threshold schedules within the bounded structural class. The alternatives share their latent treatment design and measurement channel, while opposite perturbations change the causal mean at the evaluation dose. In the inverse regimes, normalized filtered Jacobi packets preserve the perturbation’s pointwise height and cancel low-order moments under the polynomial design weight. Their localization and regularity bounds control model membership, and Gaussian convolution attenuates the resulting change in the observed outcome-marked distribution. Theorem 5 supplies target separation at the frontier order together with a uniform total-variation bound for the complete observed sample under this calibration. This comparison yields lower bounds for both estimation and honest connected-interval length over the same calibrated class used for achievability.

The closest causal measurement-error comparison is the contaminated-treatment dose-response estimator of Huang et al. (2023). Under the conditions of their arXiv version 2, Theorem 4.4, its fixed-dose expansion has bias orders $h^2+h_0^2$ and stochastic remainder $O_p\{N^{-1/2}/\min (e(h),e(h_0))\}$, where $e(b)=b^{1/2}\exp (-b^{-r}/\gamma )$ in the supersmooth case. Our contribution is a matched minimax characterization of absolute risk and honest interval length under explicit polynomial design envelopes satisfying $c_{\mathrm{lo}}\le (\kappa +1)2^{\kappa }\le c_{\mathrm{hi}}$, uniform over the known noise scale. The direct endpoint shares its rate exponent with the degenerate-design Gaussian-response regression benchmark of Gaïffas (2005), as made precise in Proposition 2. Gaussian attenuation connects the intermediate branch to errors-in-variables regression (Fan et al., 1993), while compact-support density deconvolution provides a precedent for iterated-logarithm-over-logarithm localization (Meister, 2007). These comparisons concern their respective experiments, restrictions, and losses. Under the same calibrated envelopes, the present characterization places the three mechanisms within one causal model and couples their estimation orders to honest uncertainty about the same scalar target.

The paper proceeds through related work, setup and identification, and the optimal risk and interval-length characterization. It then presents the observable estimator and honest intervals, followed by the observed-law lower bounds. Discussion and limitations and future work follow; the appendices develop identification and error certificates, weighted packets and the observed-experiment converse, and proofs of the main results.

Related work

The closest causal comparison is the contaminated-treatment dose-response estimator of Huang et al. (2023). Under their positivity, smoothing, sieve, and supersmooth-error conditions, the fixed-dose expansion in arXiv version 2, Theorem 4.4 and equation (21) has bias orders $$h^2+h_0^2$$ and stochastic remainder $$O_p\left\{\frac{N^{-1/2}}{\min (e(h),e(h_0))}\right\},e(b)=b^{1/2}\exp (-b^{-r}/\gamma ),$$ where $h,h_0$ are bandwidths, $N$ is sample size, $r$ is the supersmooth exponent, and $\gamma$ is the error-tail constant; the normal limit also uses the theorem’s variance lower bound (Huang et al., 2023). The present numerical bounds and schedule-level exchangeability specify a different structural class. Within that class, the delivered advance is a minimax characterization uniform over the known noise scale, allowing polynomially weak dose support and matching the absolute-risk order with honest connected-interval length.

The error-free endpoint connects to pointwise regression with a degenerate design. For sample size $n$, mean smoothness exponent $\beta$, and design-decay exponent $\kappa$, our direct rate is $n^{-\beta /(2\beta +\kappa +1)}$, as stated in Proposition 2. Definition 2 of Gaïffas (2005) specifies bounded local polynomial-approximation classes, and Theorem 1 characterizes minimax pointwise $L^p$ risk in direct random-design regression with independent Gaussian response errors. Under its locally symmetric, regularly varying design and regularly varying approximation modulus, the rate combines a power of sample size with a slowly varying factor. Pure-power smoothness and design behavior yield the same exponent as our direct endpoint (Gaïffas, 2005). The connection is an exponent comparison across experiments: our target is a population causal mean with bounded outcomes and stratum-specific design envelopes.

The intermediate inverse regime is related to the regression deconvolution analysis of Fan et al. (1993). Their Sections 3–4 establish estimation rates and corresponding lower bounds for errors-in-variables regression, including pointwise squared-error risk, under positive-design, smoothness, moment, and characteristic-function conditions. In the supersmooth case, exponential attenuation of high frequencies leads to logarithmic estimation rates; Gaussian errors furnish the quadratic-exponential example (Fan et al., 1993). Our intermediate branch uses this Fourier inversion mechanism within the weak-design causal class. The comparison therefore concerns how Gaussian attenuation controls localization, with each rate evaluated under its own design restrictions and loss.

Compact support supplies another relevant mechanism. Meister (2007) studies density deconvolution under mean integrated squared error over Sobolev classes, exploiting support information and knowledge of the error characteristic function on a neighborhood of the origin. For source Sobolev smoothness $s$, polynomial continuation yields the vanishing mean integrated squared error order $\{(\log \log n)/(\log n){\}}^{2s}$. The matching lower bound concerns the joint density-and-error classes of that experiment and carries additional restrictions on density smoothness, the Sobolev radius, and regularity of the local error transform. Our compact-support branch instead addresses pointwise causal evaluation under expected absolute loss, with the Gaussian error law known and the latent design constrained by polynomial envelopes.

A complementary compact-support result is the spline-assisted regression estimator of Jiang et al. (2026). Under their spline-order, dimension-growth, knot-spacing, bounded-density, compact-support, mean-smoothness, and approximation conditions, together with bounded score-derivative and second-moment operators and the locally unique nonsingular population root in (D5), Theorem 2 gives $$\underset{x}{\sup }|\hat{m}(x)-m(x)|=O_p\left((n{h}_b{)}^{-1/2}+h_b^q\right),E\{\hat{m}(x)-m(x){\}}^2=O\left(h_b^{2q}+(n{h}_b{)}^{-1}\right).$$ These algebraic spline bounds are conditional on the source experiment’s operator and nonsingularity structure (Jiang et al., 2026). In particular, their increasing-dimensional spline system must retain the bounded score operators and locally unique nonsingular population root required by (D5). The measurable polynomial envelopes and Hölder mean restriction used here do not impose that increasing-dimensional operator condition uniformly over the model class. The present result instead gives a same-class minimax characterization of pointwise absolute error and honest interval length, uniform over the maintained Gaussian noise scale; its loss and uniformity quantifiers therefore differ from the source theorem.

The interval construction connects to two strands of nonparametric inference. Armstrong et al. (2020) constructs bias-aware intervals for scalar functionals in direct regression, using a specified smoothness bound to calibrate coverage and assess interval length. Our construction likewise incorporates a uniform bias bound, together with an observable estimation-error certificate. In errors-in-variables regression, Kato et al. (2019) obtains asymptotically valid simultaneous bands through deconvolution and the multiplier bootstrap. Their result assumes ordinary-smooth measurement error, positive design and variance on the reporting interval, moment and regularity conditions, and bandwidth and auxiliary error-sample growth restrictions that include undersmoothing. Alongside these results, Theorem 3 supplies finite-sample coverage for a connected interval targeting one causal mean, while Theorem 1 characterizes its optimal worst-law expected length.

Continuous-treatment causal methods provide the broader identification and estimation context. Propensity-function adjustment identifies population treatment responses under ignorability and positivity (Imai et al., 2004), while doubly robust smoothing combines outcome regression and treatment-density estimation for dose-response estimation (Kennedy et al., 2017). Vanishing-noise regression provides a separate precedent for studying rates jointly in sample size and noise scale: Durot et al. (2026) analyzes monotone-function estimation in shuffled and unlinked experiments under integrated absolute-error criteria and the respective sampling and error conditions. These connections motivate treating measurement precision as part of the statistical characterization.

Under the envelope calibration $c_{\mathrm{lo}}\le (\kappa +1)2^{\kappa }\le c_{\mathrm{hi}}$, the contribution is a same-class characterization of minimax pointwise expected absolute error and minimum worst-law expected length of honest connected intervals, as defined in Definitions 5 and 6. For this calibrated class, fixed public $K$, exponents in their declared ranges, and any noncoverage probability $\alpha \in (0,1/2)$, Theorem 1 gives matching orders uniformly over the maintained Gaussian error range $\sigma$ in $[0,1/4]$. The observable construction and the finite-sample $(1-\alpha )$-coverage guarantee in Theorems 2 and 3, together with the observed-law converse in Theorem 5 under this calibration, place direct estimation, Gaussian Fourier inversion, and compact-support estimation within one causal experiment and one set of model restrictions.

The Gaiffas result supplies a separate direct-regression benchmark for the error-free deduction in Proposition 2; it complements the frontier’s own same-class zero-noise conclusion.

Setup and assumptions

Generic constants $c,C>0$ may depend on the fixed public class parameters; dependence on sample size or error scale is displayed explicitly. Absolute value denotes the scalar norm, and $\Vert f{\Vert }_{\infty }$ denotes the uniform norm of a function on its stated domain. The notation $u\asymp v$ means that $u/v$ is bounded above and below by positive constants uniformly over the stated index range. Stratum indices range over the two strata defined below, sample indices run from $1$ to the sample size $n$, and integer indices are nonnegative unless a different range is specified.

The structural coordinates distinguish treatment assignment, measurement error, and potential outcomes. The set of observed strata is $\mathcal{X}$, with $\mathcal{X}=\{0,1\}$, and $X$ is the observed stratum coordinate. We write $A$ for the latent dose, $Z$ for the standardized measurement error, $Y$ for the realized outcome, and $U$ for an auxiliary rank coordinate used in the lower-bound subclass. Throughout, the smoothness exponent, design-decay exponent, and public error scale are fixed at $\beta \in (0,1]$, $\kappa \in [0,2]$, and $\sigma \in [0,1/4]$, respectively.

Potential outcomes are represented by a random schedule on a fixed public measurable space $(\mathcal{S},\mathcal{E})$. Its evaluation map $e$ assigns an outcome to a schedule and a dose. The following definition makes both schedule-level independence and evaluation at a random dose measurable statements.

Write $Y(\cdot )$ for the random potential-outcome schedule and $P$ for the structural probability law of $(X,A,Z,Y(\cdot ),Y,U)$ on the corresponding coordinate spaces. The stratum probability $p_x$ is $p_x=P(X=x)$. The contaminated dose $W$ and the centered coordinates are defined by $$W=A+\sigma Z,a_0=\frac{1}{2},t=A-a_0,V=W-a_0=t+\sigma Z.$$ Here $a_0$ is the prespecified evaluation dose, $t$ is the centered latent dose, and $V$ is the centered observed dose. For each stratum with positive probability, $g_x$ denotes a Lebesgue density of the conditional law of $t$ given $X=x$; its existence and bounds are imposed below.

The population causal target $\theta (P)$ is the mean potential outcome at the evaluation dose: $$\theta (P)=E_P[Y(a_0)].$$ The boundedness condition below ensures that this expectation exists. To describe its stratum-specific components, define the conditional potential mean ${\mu }_x(a)$ before imposing restrictions on treatment assignment or mean smoothness.

This definition separates the causal mean from the distribution of assigned doses. We first restrict the stratum probabilities and the latent-dose distribution so that both strata contribute and the amount of dose support near the evaluation point is controlled.

The finite-stratum overlap condition gives each observed stratum a probability bounded away from zero, uniformly over the structural laws under consideration (Huang et al., 2023). The public constant $p_{\min }$ fixes that common lower bound. Treatment assignment is then restricted to a common compact interval.

Compact continuous-treatment support fixes the dose domain on which potential outcomes and their means are evaluated (Huang et al., 2023). Centering this interval at $a_0$ gives latent support $[-1/2,1/2]$. The next condition specifies how the conditional density behaves throughout that centered interval.

Polynomially degenerate random design describes declining dose support near the evaluation point through a fixed density envelope (Gaïffas, 2005). For $\kappa >0$, the upper envelope tends to zero as the latent dose approaches $a_0$; for $\kappa =0$, the density is bounded above and below by positive constants almost everywhere. The public constants $c_{\mathrm{lo}}$ and $c_{\mathrm{hi}}$ keep this comparison uniform across strata and structural laws.

Mean regularity supplies the corresponding control on changes in the causal response. For a real-valued function $f$ on $[0,1]$, define the Hölder increment seminorm and its radius-one class by $$[f{]}_{\beta }=\underset{\begin{array}{c}a,a^{\prime }\in [0,1]\\ a\ne a^{\prime }\end{array}}{\sup }\frac{|f(a)-f(a^{\prime })|}{|a-a^{\prime }{|}^{\beta }},{\mathcal{H}}^{\beta }([0,1])=\{f:[0,1]\to \mathbb{R}:[f{]}_{\beta }\le 1\}.$$ In particular, $[{\mu }_x{]}_{\beta }$ measures the regularity of the stratum-specific conditional potential mean.

The Hölder smoothness condition controls the approximation of a pointwise mean by means at nearby doses, as in pointwise estimation under a degenerate design (Gaïffas, 2005). Here the exponent and increment radius are fixed across the class. A separate restriction records the common range of these means.

This explicit mean-range restriction is specific to this analysis and places all conditional causal means on the unit outcome scale. It also fixes the range used by the threshold subclass introduced below. The observation channel is governed by the following two restrictions.

The Gaussian classical measurement error condition specifies the convolution law associated with the contaminated dose (Fan et al., 1993). Together with the known public scale $\sigma$, it fixes the distribution of the additive error $\sigma Z$, including the error-free case $\sigma =0$. Independence specifies its relation to the structural coordinates.

Classical independent treatment measurement error makes the same additive noise law applicable across strata, latent doses, and potential-outcome schedules (Huang et al., 2023). The causal interpretation of the latent response is supplied by schedule-level conditional exchangeability, bounded outcomes, and consistency.

Conditional exchangeability for individual potential outcomes is used in average dose-response estimation under measurement error (Huang et al., 2023). In our model, functional conditional exchangeability requires the entire measurable potential-outcome schedule to be independent of dose assignment within each observed stratum. Its schedule-level formulation accommodates evaluation at the realized latent dose through the jointly measurable map in Definition 1. Outcome boundedness provides a common scale for these evaluations.

The bounded potential outcomes condition ensures integrability at every fixed dose and fixes the unit scale of the causal target (Huang et al., 2023). It also implies the conditional mean range stated in Assumption 5. Consistency connects the potential-outcome schedule to the realized outcome.

Continuous-treatment consistency assigns the observed outcome to the potential outcome at the actual latent dose (Huang et al., 2023). Together, Assumptions 8 and 10 give the conditional potential mean its causal role in the latent-dose experiment.

For the observed-law lower bounds, we use a subclass with explicit rank-generated schedules. The auxiliary coordinate $U$ provides a common rank across doses; the next restrictions specify its conditional distribution and relation to treatment assignment.

The conditional uniform rank law is specific to this analysis and supplies a unit-interval rank for constructing binary schedules with prescribed conditional means. Its relation to the latent dose is fixed separately.

Conditional rank–dose independence is specific to this analysis and permits rank-generated schedules to satisfy the maintained conditional exchangeability restriction. The schedule construction uses the same rank at every dose.

The structural threshold restriction is specific to this analysis and constructs an entire schedule from the conditional mean function and one rank. Its simultaneous statement ties all dose evaluations to a single measurable schedule. Under the conditional uniform rank law, each threshold has the prescribed conditional mean. The resulting subclass also satisfies the following outcome restriction.

Average dose-response estimation under measurement error provides a related causal setting (Huang et al., 2023). Binary potential outcomes provide the outcome specification for the lower-bound witnesses within our bounded-outcome causal model. The threshold construction satisfies this restriction directly.

We now collect the defining restrictions into the bounded-outcome structural model class ${\mathcal{P}}_{K,\beta ,\kappa ,\sigma }$. The public vector $K$ records the common stratum-overlap and density-envelope constants.

This class fixes the structural restrictions under which causal estimation is evaluated. Its lower-bound subclass supplies explicit binary threshold laws satisfying those same restrictions.

An observed record $O$ consists of $O=(X,W,Y)$. For a sample of size $n$, write $O_i=(X_i,W_i,Y_i)$, $i=0,\dots ,n-1$, for the corresponding records; this zero-based indexing is the one used by the modulo-three sample blocks of the estimator. Where a statement lists the sample as $O_1,\dots ,O_n$, it denotes the same $n$ records. The sampling restriction determines how their common observed law generates the statistical experiment.

Independent and identically distributed sampling specifies repeated observations from the same contaminated-dose experiment (Huang et al., 2023). The induced sample law $Q_P^n$ is the experiment available to estimators and confidence intervals.

Causal identification must account for both measurement error and the declining density near the target dose. Proposition 1 establishes that, for common public class parameters and path space, equality of the observed-record laws determines the conditional potential means throughout the dose interval and hence the population causal target.

The shrinking-neighborhood conclusion explains how the target remains identified under weak dose support. Every displayed neighborhood has positive conditional latent mass, and its mean-measure to dose-measure ratio converges to the stratum-specific causal mean at $a_0$. Thus the pointwise target has a population interpretation through local averages even when the density envelope declines to zero near that dose. The proposition also identifies the entire conditional mean function, while its final decomposition combines the two stratum-specific target values using the identified stratum probabilities. The realized-dose identity and Gaussian convolution argument supporting these conclusions are presented in Lemmas 1 and 2; the positive neighborhood masses and the convergence of the local ratio are proved directly in steps 3 and 4 of the proof of Proposition 1.

Optimal estimation and interval length

Identification in Proposition 1 makes the population causal mean a well-defined target of the observed experiment. We now characterize the precision attainable from that experiment as sample size and measurement precision vary. The first criterion is the minimax expected absolute error $R_n(\sigma )$, evaluated over the structural class in Definition 3.

For uncertainty quantification, fix a noncoverage probability $\alpha \in (0,1/2)$, corresponding to coverage of at least $1-\alpha$ under every law in the structural class. The second criterion, $H_n(\sigma )$, measures the smallest worst-law expected length among connected intervals satisfying this uniform coverage requirement. The separate finite-sample upper guarantee in Theorem 3 holds for the wider range $\alpha \in (0,1)$; the restriction $\alpha <1/2$ is used by our two-point argument for the honest-length lower bound, which requires $1-2\alpha$ to be positive; we do not claim that the frontier fails when $\alpha \ge 1/2$.

Both criteria are governed by a common rate ${\rho }_{n,\sigma }$, obtained by raising the localization scale $b_{n,\sigma }$ to the mean-smoothness exponent. The effective exponent $d$ combines smoothness and local design decay, while the logarithmic sample-size quantity $L_n$ determines the second measurement-error threshold. Their definitions distinguish three ranges of measurement precision.

The main characterization, Theorem 1, gives matching bounds for estimation error and honest interval length throughout the stated error-scale range. It also compares the neighboring formulas across multiplicative neighborhoods of both thresholds, making the rate characterization uniform through the transitions.

The envelope calibration in Theorem 1 has a concrete role: the density $(\kappa +1)2^{\kappa }|t{|}^{\kappa }$ integrates to one on $[-1/2,1/2]$, and bracketing its coefficient places the explicit lower-bound design inside the class. Thus the condition certifies a nonempty class containing the witnesses used for the converse.

Under the envelope calibration $c_{\mathrm{lo}}\le (\kappa +1)2^{\kappa }\le c_{\mathrm{hi}}$, the three branches express the interaction between local dose support and measurement precision. In the direct regime, the measurement-error scale is at most the localization scale available from an error-free dose record. Local design decay enters through $d=2\beta +\kappa +1$, and the resulting rate is $n^{-\beta /d}$. In the intermediate Gaussian regime, recovering a localized mean from the contaminated dose requires Gaussian inversion, with attainable localization scale $F_n(\sigma )$. For larger error scales, known compact latent support enables polynomial localization: inverse polynomial degree is the localization width, Gaussian inversion increases the inverse weight’s second moment with degree, and the largest usable degree has order $L_n/\log (e+{\sigma }^2{L}_n)$. Balancing these effects gives the polynomial localization scale $P_n(\sigma )$ and explains the transition from the Fourier branch. The detailed public certificate appears in Lemma 5. For the calibrated class, raising each localization scale to the smoothness exponent gives both the estimation rate and the honest-length rate.

Table 1 displays these branches and their thresholds. The scales $D_n$, $F_n(\sigma )$, and $P_n(\sigma )$ are those defined in Theorem 1; each row gives the rate up to the uniform constants in that result.

The transition comparisons give the thresholds a precise interpretation. For each fixed multiplicative factor $W\ge 1$, the direct and intermediate scales are comparable throughout $[a_n/W,W{a}_n]$ once the sample size is sufficiently large. Likewise, the intermediate and compact-support scales are comparable throughout $[b_n/W,W{b}_n]$. Thus neighboring descriptions yield the same order of uncertainty throughout each transition window. The comparison constants may depend on $W$ and the fixed class parameters, while remaining uniform over the public potential-outcome path spaces.

For every fixed positive measurement-error scale in the stated range, increasing sample size eventually places the experiment in the compact-support regime. Both minimax error and minimum honest expected length then have order $(\log \log n/\log n{)}^{\beta }$, with comparison constants that may depend on the fixed error scale. The full uniform characterization also describes sequences of improving measurement precision: attaining the direct-regime order requires an error scale at most of order $n^{-1/d}$, including its fixed multiplicative transition neighborhood.

The error-free endpoint follows directly from this characterization. The next result states that endpoint for both criteria and places its estimation rate alongside a direct regression benchmark with the same local design-decay exponent.

Proposition 2 identifies the precision available when the latent dose is observed exactly. Its exponent reflects both mean smoothness and the declining frequency of observations near the evaluation dose. The comparison with Gaiffas’s benchmark is a comparison of minimax absolute-error rates for the displayed function class and fixed benchmark parameters (Gaïffas, 2005). At $\beta =1$, that benchmark retains approximation by polynomials of degree at most one. The honest-length conclusion for the causal experiment is supplied by the first part of Proposition 2. Together, these conclusions provide an error-free reference for interpreting the measurement-error branches.

The upper and lower bounds have complementary statistical roles. Observable localization produces estimates of the stratum-specific causal means from contaminated-dose and outcome records; the construction and its honest intervals are given in Theorems 2 and 3. The converse compares structural laws whose causal means are separated while their induced observed laws remain difficult to distinguish, as stated in Theorem 5. Working with the observed experiment makes the lower bound apply to every estimator and honest connected interval in the criteria above.

The dependency roadmap is short. Identification in Proposition 1 fixes the observed-law target. The public certificate in Lemma 4 turns each inverse weight into a risk and interval-radius bound; the dictionary comparisons in Lemma 5 select entries at the three frontier scales. Finally, the observed-law witnesses in Theorem 5 convert packet cancellation into matching testing lower bounds. Detailed algebra and witness coordinates are deferred to the two technical appendices.

The weighted-packet construction supporting that converse is developed in Theorem 4. Its mathematical ingredients include filtered Jacobi localization (Petrushev et al., 2005), the corresponding local difference estimate (Kyriazis et al., 2008), Jacobi orthogonality and squared norms (Standards et al., ), and the positive-real-axis gamma-ratio asymptotic (Standards et al., ). These ingredients supply localization and moment cancellation for the observed-law comparison; their detailed construction belongs with the auxiliary results in the appendix.

Observable estimation and honest intervals

The observable estimation layer uses the public class constants, smoothness and design-decay indices, and Gaussian error scale to choose a localization weight before evaluating the sample. Exact Gaussian inversion turns that latent weight into an observable function of the contaminated dose. A finite dictionary then supplies a single procedure whose expected absolute error attains the rate in Definition 7, uniformly over the stated error scales.

Inverse weights and public scores

Let $q$ be a public nonnegative weight on the centered latent-dose interval, and let ${\ell }_q$ be its observable Gaussian inverse. The inverse identity requires that averaging the observable inverse over the measurement error recover the latent weight at every dose in the support. The following definition groups this identity with the integrability conditions and the public quantities used to evaluate a candidate.

The design-weighted moment $I_{\kappa }(q)$ determines the public denominator bound $b_q(K)$. Localization is measured by the bias quantity $B_q(K)$, while the inverse-weight second-moment quantity $V_q(K)$, equal to $c_{\mathrm{hi}}/16$ times the factor-$16$ reference moment $V_q^{(16)}$, measures the sampling cost of Gaussian inversion. Their combination, the public expected-error score $A_q(K)$, also accounts for estimation of the stratum probabilities. All these quantities depend on the candidate and public parameters. The factor-$16$ convention is related to the class-dependent second moment by the explicit conversion in Definition 8.

For interpretation, define the localized population ratio $m_{x,q}$ by $$m_{x,q}=\frac{{\int }_{-1/2}^{1/2}{\mu }_x(a_0+t)q(t)g_x(t)dt}{{\int }_{-1/2}^{1/2}q(t)g_x(t)dt},x\in \mathcal{X}.$$ This ratio averages the conditional causal mean over doses emphasized by the weight. The positive weighted moment and lower design envelope give a positive denominator. Exact inversion supplies observable weighted moments for estimating this ratio; Lemma 4 connects the resulting estimator to the public score.

Two explicit families furnish the dictionary comparisons. The polynomial localization weight $q_m^{\mathrm{M}}$, indexed by a positive odd integer $m$, has the inverse-heat companion ${\ell }_m^{\mathrm{M}}$. Its polynomial form makes the inverse a finite derivative sum.

The Fourier family instead uses the localization function $Q$ and a bandwidth $h$. Its latent weight $q_h^{\mathrm{F}}$ is paired with the Gaussian inverse ${\ell }_h^{\mathrm{F}}$ through a Fourier multiplier. Both families satisfy the same exact inversion requirement.

Public selection and the observable procedure

The finite public dictionary ${\mathcal{D}}_{n,\sigma }$ combines these two families with a constant entry. Each entry carries its inverse and its score, so selection compares the same public criterion across localization methods.

The constant entry ensures that the dictionary is nonempty. A fixed order resolves ties and defines the selected entry $\hat{q}$ uniquely, including when differently tagged entries have the same latent weight.

Because the scores and order use public inputs, the selected entry is determined by the sample size and declared parameters. The sample supplies the weighted moments used after selection. To state those formulas, partition the records into deterministic modulo-three blocks ${\mathcal{I}}_r$. The three blocks separately estimate the stratum frequency ${\hat{p}}_x$, the weighted denominator ${\hat{D}}_x(q)$, and the weighted outcome numerator ${\hat{M}}_x(q)$.

The stratum-specific ratio estimate ${\hat{\mu }}_x(q)$ uses a public denominator floor and projection onto the outcome range. Combining these ratios with estimated stratum frequencies gives the weight-specific causal-mean estimate ${\hat{\theta }}_q$. The following definition supplies the intermediate formulas and the empty-block branch.