Introduction

This paper establishes a sharp pointwise precision benchmark for conditional average treatment effect estimation when responses are observed on a smaller collection of records than covariates and treatment assignments. We consider $n\ge 2$, the labeled-record count, and $m\ge 0$, the independent outcome-free covariate-treatment record count. For the specified potential-outcome class in Definition 3, with known uniform covariate design and the public parameter restrictions stated below, the minimax expected absolute error is $$\underset{\text{minimax pointwise absolute-error risk}}{\underbrace{R(n,m)}}\asymp \max \left\{n^{-\gamma /(2\gamma +d)},[n(n+m){]}^{-\gamma /(2\gamma +d+\gamma d/(\alpha +\beta ))}\right\},$$ where $d$ is the covariate dimension, and $\alpha ,\beta ,\gamma$ are the Hölder smoothness orders of the propensity, control response mean, and causal contrast, respectively. The target is the conditional causal contrast at the cube’s center. The comparison holds uniformly over both record counts, with constants depending on the fixed public class parameters. A fully specified decision based on the observed records attains this order, as established in Theorem 2.

The model fixes the covariate distribution to be uniform on $[0,1{]}^d$, with $d\ge 1$, and assumes conditional treatment exchangeability. Treatment and potential responses are binary. For a fixed public $\epsilon \in (0,1/2)$, the propensity lies in $[\epsilon ,1-\epsilon ]$, and both potential-response means lie in $[0,1]$ throughout the cube. The public smoothness orders satisfy $\alpha ,\beta \in (0,1]$ and $\gamma \in [1,3]$, with $L$, the common public Hölder radius, satisfying $L>1/2$. These restrictions define the primitive law class in Definition 3; the independent, common-population record blocks follow Assumption 9. Admissible conditional-margin witnesses satisfy the restrictions over the whole cube. Their continuous contrast, together with full-support design and the identifying assumptions, determines the point target uniquely from the observed law.

The distinction between record types has a direct acquisition interpretation. Response ascertainment through chart review can yield labeled records containing covariates, treatment, and the true response, while routine records supply covariates and treatment. Surrogate-assisted treatment-effect estimation provides a related electronic-record setting (Cheng et al., 2021). Within the present sampling model, adding a new independent labeled record changes $(n,m)$ to $(n+1,m)$. Labeling one already-counted outcome-free record changes the composition to $(n+1,m-1)$ at fixed total count; if the record index is fixed in advance, or chosen by a randomization independent of all records, the resulting dataset follows the stated sampling law at $(n+1,m-1)$. Adding a new outcome-free record changes $(n,m)$ to $(n,m+1)$. The sharp risk formula benchmarks these sample compositions under the stated independent-sampling assumptions.

The estimator combines a coarse local polynomial fit with a finer projection correction. It splits the original records into independent outcome and treatment roles, whose sizes are proportional to the labeled and total treatment-record counts. A rectangular average across these roles corrects the local estimating moments for the unknown nuisance functions. An empirical eigenvalue guard and clipping produce a finite Borel decision for every dataset. Under the stated model conditions, Theorem 1 bounds its risk by effect approximation, nuisance approximation, localized response sampling, and rectangular sampling terms. Public tuning balances these terms to attain the sharp rate.

The matching lower bound retains the original fixed-size observation experiment. The construction in Theorem 4 couples propensity and response perturbations while preserving identical auxiliary-only mixture laws across hypotheses. Its information bound supplies the nuisance converse in Lemma 6 over the range where the product-of-counts term governs precision. The supplied-propensity comparison in Theorem 3 establishes the labeled-sample order $n^{-\gamma /(2\gamma +d)}$ over the same primitive class and every auxiliary count. Together, these lower bounds and the attaining estimator give the uniform minimax characterization, including randomized decisions.

The characterization distinguishes strict improvement from attainment of the supplied-propensity order. Write $S=\alpha +\beta$, the combined nuisance smoothness, and let $S_{\mathrm{crit}}$ denote the smoothness threshold and $q_{\ast }$ the total-record growth exponent: $$S_{\mathrm{crit}}=\frac{\gamma d}{2\gamma +d},q_{\ast }=\frac{\gamma d}{S(2\gamma +d)}.$$ At and above the threshold, labeled records alone attain the supplied-propensity order. Below it, attainment along an auxiliary-size sequence requires total treatment-record counts to remain at least a positive constant times $n^{q_{\ast }}$. Below the smoothness threshold, strict asymptotic improvement over labeled records alone occurs exactly when the auxiliary-to-labeled ratio diverges. Proposition 1 also translates a target absolute-error scale into simultaneous requirements on the labeled count and the product of labeled and total counts. These are risk requirements up to parameter-dependent constants, with exact equivalences for the displayed rate.

The closest supervised comparison is Kennedy et al. (2024). At zero auxiliary records, our rate has the same numerical exponents and smoothness cutoff, as recorded in Lemma 7. Their statistical guarantees concern their bounded-density model and stated attainment conditions; the lower bound for our known-uniform class combines the original-record construction in Theorem 4 and Lemma 6, which covers smoothness below $S_{\mathrm{crit}}$ with total count at most $n^{q_{\ast }}$, with the labeled-sample floor $n^{-\gamma /(2\gamma +d)}$ from the constant-nuisance subfamily in Theorem 3. Semi-supervised analyses of average and quantile treatment effects characterize nuisance-dependent expansions and inference for marginal targets (Chakrabortty et al., 2022). Our contribution characterizes worst-case pointwise absolute error and its acquisition requirements for the specified conditional target. The projection construction connects local residualized regression (Kennedy, 2023) with finite-rank second-order correction (Robins et al., 2009).

The exposition proceeds through related work, setup and assumptions, the estimator construction, sharp precision and sample requirements, and discussion and limitations. The upper-bound argument is organized in Section A, the original-record converse in Section B, and the acquisition algebra in Section C.

Related work

The closest comparison is Kennedy et al. (2024), who characterize supervised pointwise CATE estimation under Hölder smoothness. Write $n$ for the labeled sample size, $d$ for the covariate dimension, and $\alpha$, $\beta$, and $\gamma$ for the propensity, control-mean, and treatment-effect smoothness orders. With $s_{\mathrm{pub}}=(\alpha +\beta )/2$, their absolute-error exponents are $$\frac{1}{1+d/(2\gamma )+d/(4s_{\mathrm{pub}})}\text{when}{s}_{\mathrm{pub}}<\frac{d/4}{1+d/(2\gamma )},\frac{\gamma }{2\gamma +d}\text{otherwise}.$$ Published attainment requires their matrix regularity, smooth nuisance-estimation errors, boundedness conditions, and control of the third-order remainder involving covariate-distribution estimation. Their localized second-order projection correction balances effect approximation, nuisance approximation, and sampling variation.

The algebraic comparison appears in Lemma 7. It identifies the supervised exponents obtained by specializing the sample-size expression to zero auxiliary records. Its interpretation follows the published source model, which permits a bounded covariate density and uses a construction-specific design in the lower-bound argument. The present potential-outcome law class with known uniform design, specified in Definition 3, fixes the covariate distribution throughout. Within that class, Theorem 4 and Lemma 6 supply the original-record testing argument. These results establish the lower bound in the experiment actually observed by the decision maker, while the published benchmark supplies the numerical supervised comparison.

This distinction becomes consequential when information arrives in unequal amounts. The sampling experiment combines $n$ labeled records with $m$ independent outcome-free covariate-treatment records; its sampling law is given in Assumption 9. The pointwise absolute-error characterization in Theorem 2 retains both record counts, and Theorem 3 identifies the precision attainable when the propensity function is supplied. Together, these results connect the amount of treatment information to the accuracy of a conditional causal contrast under the paper’s fixed design, overlap, and smoothness conditions.

Related semi-supervised treatment-effect analyses study population functionals through asymptotic expansions. Chakrabortty et al. (2022) separate labeled residual variation from averaging over the enlarged covariate sample for average treatment effect estimation. Their corollary addresses a vanishing labeled fraction with a correctly specified propensity model, retaining explicit nuisance remainders; asymptotic normality follows under the stated remainder conditions. The corresponding quantile treatment effect expansions additionally account for the density at the target quantile (Chakrabortty et al., 2022). These expansions explain robustness and efficiency for marginal causal functionals. The present risk characterization instead evaluates a conditional contrast at a fixed interior profile and quantifies worst-case absolute error over the specified Hölder class.

The information carried by auxiliary records also distinguishes the surrogate-assisted setting of Cheng et al. (2021). Their records include outcome-predictive surrogate variables, potentially measured after treatment, alongside baseline covariates and treatment. An imputation-based ATE estimator exploits these variables, with robustness to imputation-model misspecification and local efficiency under their ideal semi-supervised model; perturbation resampling supports inference. Their electronic health records application illustrates the role of surrogate information when chart-reviewed outcomes are scarce. Here, auxiliary observations contribute covariate-treatment information, and their value is characterized through the pointwise risk and sample requirements in Theorem 2 and Proposition 1.

A further comparison concerns the statistical criterion used to measure precision. Kato (2025) derive two-sample ATE efficiency bounds for regular estimators and establish attainment through cross-fitted estimation under nuisance consistency and product-rate conditions. Their auxiliary sample contains covariates, and their analysis allows distinct labeled and auxiliary covariate distributions subject to support conditions. With coincident distributions and the stated evaluation weighting, the efficiency bound separates outcome residual variation from variation in the conditional effect across covariate profiles. Those conclusions characterize asymptotic variance at a fixed positive sampling fraction. The present minimax criterion, defined in Definition 5, measures pointwise absolute error across the primitive class and accommodates every auxiliary-record count in the stated experiment.

Supplied-predictor conditional estimation offers another route for using auxiliary information. Sui et al. (2026) combine labeled observations, unlabeled covariates, and a supplied predictor to estimate conditional moment targets. Their Sobolev-equivalent reproducing kernel Hilbert space conditions support pointwise error bounds and confidence intervals, with prediction and residual components determining variance. Structure-adaptive CATE estimation uses different restrictions: Kim (2026) give a pointwise squared-error result under bounded ambient-RKHS and overlap conditions when the contrast lies in the designated lower-complexity subspace; their broader analysis also treats integrated squared error, source conditions, known lower-dimensional representations, and model selection. Those results use supervised outcomes and RKHS/Sobolev structure. Our experiment instead fixes a Hölder class, measures pointwise absolute error, and studies the extra information supplied by outcome-free covariate-treatment records.

Methodologically, the projection correction belongs to the higher-order influence-function tradition. Finite-rank orthogonal projections provide a tractable representation of second-order corrections in nonlinear functional estimation (Robins et al., 2009). Orthogonal regression supplies the conditional-effect estimating structure: residualized treatment and outcome regression underlie the R-learner (Nie et al., 2021), and local polynomial double-residual regression connects this structure to pointwise smoothness-based estimation (Kennedy, 2023). The rectangular projection estimator in Algorithm 2 and Definition 11 combines localization with independent treatment and outcome roles. Its upper bound in Theorem 1 makes the separate sampling contributions explicit, linking the projection architecture to the unequal information available from labeled and outcome-free records.

Setup and assumptions

Throughout, $c$ and $C$ denote finite positive constants whose values may change between appearances. At fixed public parameters, these constants are independent of the sample sizes. We write $a\lesssim b$ when $a\le Cb$, and $a\asymp b$ when both $a\lesssim b$ and $b\lesssim a$. Vector norms are Euclidean unless otherwise indicated. We use $\mathbb{N}={\mathbb{N}}_0=\{0,1,\dots \}$ and write ${\mathbb{N}}_{\ge 1}$ for the positive integers. Counts and polynomial multi-indices may be zero; dimensions, resolutions, and sample-index ranges carry their displayed positive lower bounds. The symbol $N=n+m$ always denotes the total treatment-record count.

Fix the public parameters $$d\ge 1,\alpha ,\beta \in (0,1],\gamma \in [1,3],L>1/2,\epsilon \in (0,1/2).$$ Here $d$ is the covariate dimension, $\alpha ,\beta ,\gamma$ are the propensity, control-mean, and causal-contrast smoothness orders, respectively, $L$ is the common Hölder radius, and $\epsilon$ is the overlap level. These parameters remain fixed as the sample sizes vary.

Primitive variables and identifying restrictions

We use a coordinate-partial convention for Hölder regularity. For a function $f$ and a regularity order $s$, the norm $\Vert f{\Vert }_{H^s}$ controls coordinate derivatives indexed by the nonnegative-integer multi-index $\kappa$, together with their highest-order Hölder variation. The following definition specifies the convention, including at integer smoothness orders.

For positive $s$, this convention uses derivatives through order $\lceil s\rceil -1$ and a highest-order Hölder exponent of $s-\lceil s\rceil +1$. Thus integer orders use a Lipschitz condition on derivatives of one lower order. Every regularity restriction below is evaluated within the unit cube.

A full record consists of a covariate vector $X$, a binary treatment indicator $A$, and binary potential responses $Y(0)$ and $Y(1)$ under control and treatment. Let $P=\mathcal{L}(X,A,Y(0),Y(1))$ denote their primitive Borel law. The observed response $Y$ selects the potential response corresponding to the realized treatment.

The evaluation profile is the fixed interior point $x_0=(1/2,\dots ,1/2)$. We write $e_P$ for an admissible propensity witness, ${\mu }_{0,P}$ and ${\mu }_{1,P}$ for admissible control and treated conditional-mean witnesses, and ${\tau }_P$ for the canonical continuous causal contrast. Their precise admissibility requirements are collected in Definition 3. The identifying restrictions begin with the covariate design and treatment assignment.

The known uniform design fixes the distribution used for localization and integration and gives full support on the cube. It is the maintained uniform-design restriction of the bounded-density framework considered by (Kennedy et al., 2024). Treatment assignment is linked to the potential responses through conditional exchangeability.

Conditional exchangeability allows treatment-specific observed means to identify the corresponding potential-response means after conditioning on the covariates (Chakrabortty et al., 2022). Both treatment arms must also be represented throughout the evaluation domain.

Bounded overlap keeps both conditional treatment probabilities away from zero, as in the pointwise conditional-effect framework of (Kennedy et al., 2024). Here the endpoints are the public values $\epsilon$ and $1-\epsilon$, with $0<\epsilon <1/2$, and the restriction applies at every point of the cube.

The regularity conditions distinguish the two nuisance functions from the causal contrast. We first impose isotropic Hölder regularity on the propensity.

Isotropic Hölder propensity regularity controls how treatment probabilities vary across nearby covariate profiles (Kennedy et al., 2024). The order $\alpha$ is common across coordinates, and the radius $L$ bounds both the function and its Hölder variation. The control response mean has its own smoothness order.

Isotropic Hölder control-mean regularity governs variation in the baseline response surface (Kennedy et al., 2024). Allowing $\beta$ to differ from $\alpha$ separates outcome regularity from treatment-assignment regularity. The target surface is controlled directly by the following restriction.

Isotropic Hölder contrast regularity controls the local variation of the conditional treatment effect itself (Kennedy et al., 2024). The separate order $\gamma \in [1,3]$ permits a smoother causal contrast than either nuisance function and specifies the regularity used to estimate its value at $x_0$.

The binary-response model also requires pointwise probability bounds on the designated arm means. We state these restrictions explicitly because admissible witnesses are functions defined throughout the cube.

This pointwise control-response restriction is specific to the present analysis and keeps the designated control mean within the Bernoulli probability range, including its endpoints. The corresponding treated mean obeys the same numerical bounds.

The treated-response restriction is likewise specific to this analysis. Together, the two arm bounds ensure that the designated contrast takes values in $[-1,1]$ everywhere on the cube.

Membership in the primitive class is existential: a law belongs when it admits conditional-mean witnesses satisfying the stated restrictions. Let $\mathcal{M}(d,\alpha ,\beta ,\gamma ,L,\epsilon )$ denote this class, abbreviated to $\mathcal{M}$ at the fixed public parameters.

The witness formulation separates almost-sure conditional-expectation identities from restrictions imposed at every covariate profile. Under the positive smoothness orders fixed above, $e_P$, ${\mu }_{0,P}$, and ${\tau }_P$ are continuous; the identity in Definition 3 then gives continuity of ${\mu }_{1,P}$ as well.

Observed response selection, conditional exchangeability, and overlap yield $${\mathbb{E}}_P[Y\mid A=a,X]={\mu }_{a,P}(X)P\text{-almost surely},a\in \{0,1\}.$$ Consequently, the observed-record law identifies the causal contrast almost everywhere under the covariate distribution. Uniform design assigns positive probability to every nonempty relatively open subset of the cube. Continuous representatives that agree almost everywhere therefore agree everywhere, which makes ${\tau }_P(x_0)$ a uniquely defined, identified point target. The continuous representative gives substantive meaning to evaluation at a fixed profile even though conditional expectations are initially defined through almost-sure identities.

Sampling experiment and risk

Write $P^o={\mathcal{L}}_P(X,A,Y)$ for the observed-record law induced by Definition 2, and $P_{XA}={\mathcal{L}}_P(X,A)$ for the covariate-treatment marginal. For $n\ge 2$ and $m\ge 0$, the original dataset ${\mathcal{D}}_{n,m}$ consists of $n$ independent labeled records with law $P^o$ and $m$ independent outcome-free records with law $P_{XA}$: $${\mathcal{D}}_{n,m}=\left(((X_i,A_i,Y_i){)}_{i=1}^n,(({\tilde{X}}_j,{\tilde{A}}_j){)}_{j=1}^m\right).$$ The two blocks are independent. A uniform randomizer $U$ on $[0,1]$, independent of both blocks, is available to decisions. Thus the product sampling law ${\mathsf{E}}_{n,m}(P)$ in Assumption 9 is $${\mathsf{E}}_{n,m}(P)=(P^o{)}^{\otimes n}\otimes P_{XA}^{\otimes m}\otimes \mathrm{Unif}([0,1]).$$ The following condition makes this sampling experiment explicit.

Independent fixed-size labeled and outcome-free samples from a shared law provide the two-channel sampling structure used in semi-supervised causal estimation (Chakrabortty et al., 2022); the independent randomizer permits randomized decisions. In this experiment, each labeled record supplies covariates, treatment, and a response, while each outcome-free record supplies covariates and treatment. The common marginal $P_{XA}$ makes both blocks informative about treatment assignment.

A decision $T$ is a Borel real-valued function of the dataset and randomizer. Its absolute-error risk ${\mathcal{R}}_{n,m}(T,P)$ measures expected error in estimating the identified point contrast.

Absolute loss expresses precision on the scale of the conditional response difference. To assess precision uniformly across admissible primitive laws, the minimax risk $R(n,m)$ optimizes the worst-case value of this loss over all Borel decisions.

The sample sizes enter separately because response information is supplied by the labeled block and treatment-assignment information is supplied by both blocks. A complementary experiment supplies the entire admissible propensity function to the decision. Denote such a decision by $T^e$, and its minimax risk by $R^e(n,m)$.

Supplying $e_P$ gives a reference experiment in which treatment probabilities are available throughout the cube. Measurability is required in the observed data and randomizer for each fixed supplied function. Comparing Definitions 5 and 6 quantifies how the original sampling experiment approaches the precision available with supplied treatment probabilities.

Rate notation

We introduce the rate notation as a single family. Let $N=n+m$ be the total treatment-record count and $S=\alpha +\beta$ the sum of the two nuisance smoothness orders. Define the tuning denominator $\Delta$, nuisance-smoothness cutoff $S_{\mathrm{crit}}$, and total-record comparison exponent $q_{\ast }$ by $$\Delta =2\gamma +d+\frac{\gamma d}{S},S_{\mathrm{crit}}=\frac{\gamma d}{2\gamma +d},q_{\ast }=\frac{\gamma d}{S(2\gamma +d)}=\frac{S_{\mathrm{crit}}}{S}.$$ The supplied-propensity reference order $r_o(n)$ and the unequal-channel upper-rate expression $r_{\mathrm{up}}(n,m)$ are $$r_o(n)=n^{-\gamma /(2\gamma +d)},r_{\mathrm{up}}(n,m)=\max \left\{r_o(n),(nN{)}^{-\gamma /\Delta }\right\}.$$ These expressions distinguish a term indexed by the response-bearing sample size from a term indexed by the product of the labeled and total treatment-record counts. Their statistical interpretation is supplied by Theorems 1, 2, and 3.

The notation also makes the sample-size comparison transparent. Algebraically, $$(nN{)}^{-\gamma /\Delta }\le r_o(n)⟺N\ge n^{q_{\ast }}.$$ When $S\ge S_{\mathrm{crit}}$, the exponent satisfies $q_{\ast }\le 1$, so $N\ge n$ meets this comparison. For $S<S_{\mathrm{crit}}$, one has $q_{\ast }>1$, and the comparison requires total treatment-record growth beyond the labeled count. This division organizes the precision and sample-requirement statements that follow.

Estimation with outcome-free records

The estimator allocates labeled and outcome-free records to two independent statistical roles. One role supplies response information; the other supplies treatment-assignment information. A local polynomial fit estimates the contrast at the target profile $x_0$, while a finer projection corrects its empirical moments for the unknown propensity and control mean. Separate localization and projection scales allow the construction to use the smoothness of the effect and the combined smoothness of the nuisance functions.

We first specify the neighborhood associated with the localization scale $h$ and its probability measure.

The localization cube $C_h$ lies inside the covariate domain. Its uniform probability measure ${\nu }_h$ defines the inner product for both polynomial spaces below. Under Assumption 1, weighting an observed covariate by the localization density $w_h$ converts a population average into an integral under this local probability measure.

The coarse basis $r$ consists of scaled tensor-product Legendre polynomials of total degree at most two. This degree accommodates local polynomial approximation throughout the stipulated effect-smoothness range.

The vector $r$ is the coarse orthonormal basis and has $q=\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)$ coordinates. Write $r_0=r(x_0)$ for its value at the target profile. A coefficient vector $v\in {\mathbb{R}}^q$ represents the coarse polynomial $p_v(x)=r(x{)}^{\top }v$; evaluating the fitted contrast therefore amounts to evaluating its coefficient vector against $r_0$. The prescribed fine basis is ${\mathbf{b}}_J$ from the preceding definition. It has $k=q{J}^d$ coordinates and spans the already specified cellwise space $V_J$; the associated projection is defined next.

The projection kernel $K_J$ represents the orthogonal projection ${\Pi }_J$ onto $V_J$. The coarse fit operates across the entire localization cube, whereas the projection resolves nuisance variation on cells of side length $\delta$. Refining the grid from $J$ to a multiple $J^{\prime }$ of $J$ subdivides every cell, so $V_J\subseteq V_{J^{\prime }}$ and the correction space is enriched; at every resolution $J$, $V_J$ contains the coarse polynomials, so the projection reproduces them.

We next assign the original observations to independent roles under the sampling law in Assumption 9.

The concatenated treatment sequence ${\mathcal{D}}^T$ retains the covariate coordinate $X_i^T$ and treatment coordinate $A_i^T$ of each record. The outcome-role count $\ell$ is proportional to the labeled sample size, while the treatment-role count $t$ is proportional to the total treatment-record count. Their index sets, $I_L$ and $I_T$, select disjoint records. Thus additional outcome-free observations enlarge the treatment role while the response information remains tied to the labeled sample. The split also makes products of empirical averages across roles suitable for estimating products of population moments.

To interpret the response correction, write $${\eta }_P(x)={\mathbb{E}}_P[Y\mid X=x]={\mu }_{0,P}(x)+e_P(x){\tau }_P(x),$$ using the designated witnesses from Definition 3. At the population level, residualizing treatment against its propensity gives $${\mathbb{E}}_P[(A-e_P(X))Y\mid X=x]=e_P(x)\{1-e_P(x)\}{\tau }_P(x).$$ The empirical construction implements this local moment relation through a finite projection and independent roles. Its outcome projection uses the observed response, and its treatment role supplies the assignment factor in the projection correction.

The empirical outcome projection ${\hat{\eta }}_J^L$ estimates ${\Pi }_J{\eta }_P$. In the response vector ${\hat{R}}_{h,J}$, the first average estimates the treated-response moment and the rectangular average supplies its projected treatment-response correction. The raw matrix ${\hat{Q}}_{h,J}^{\mathrm{raw}}$ applies the corresponding correction to the local polynomial design. Symmetrization produces the matrix ${\hat{Q}}_{h,J}$ used for inversion.

Define the population matrix $Q_{P,h,J}$ and population response vector $\bar{R}$ by $$Q_{P,h,J}={\mathbb{E}}_P[{\hat{Q}}_{h,J}],\bar{R}={\mathbb{E}}_P[{\hat{R}}_{h,J}],$$ where expectations are taken under the original sampling experiment. These are the population counterparts of the projected local estimating equations. Polynomial reproduction links those equations to approximation of the contrast, and overlap supplies the population positivity used to control inversion; the respective statements are Lemmas 1 and 2.

For the sample decision, an eigenvalue guard supplies a criterion for inversion at the public overlap level. We use the clipping operator $${\mathrm{clip}}_{[-1,1]}(z)=\max \{-1,\min \{1,z\}\},$$ which projects a real number onto the contrast range implied by the probability-valued arm means in Definition 3. The guard level is $$g(\epsilon )=\frac{\epsilon (1-\epsilon )}{2},$$ half the population Gram margin supplied by overlap. The complete decision combines the empirical moments, guard, and clipping.

The guarded decision ${\hat{T}}_{h,J}$ evaluates the fitted polynomial when the empirical matrix meets the eigenvalue cutoff and returns zero otherwise. Clipping bounds the decision on both branches. Together, these operations permit a risk guarantee over every admissible localization scale and grid resolution, including choices with sparse local observations.

The public choices of localization scale $h_{\ast }$ and grid resolution $J_{\ast }$ balance approximation and sampling error. They determine the tuned decision ${\hat{T}}_{\mathrm{up}}$ using the sample sizes and the public class parameters.

When $S\ge \gamma$, the grid choice is $J_{\ast }=1$. When $S<\gamma$, the grid becomes finer as the localization scale shrinks, making ${\delta }^S$ comparable in order to $h^{\gamma }$. Thus nuisance approximation can use a finer spatial scale than the local effect fit. The localization choice then balances these approximation terms against the two sampling terms stated below.

For use in the decision-theoretic comparison, the same estimator is recorded as the original-record decision $T_{\ast }$.

The following guarantee gives the construction’s four statistical terms once: effect approximation $h^{\gamma }$, nuisance approximation ${\delta }^{\alpha +\beta }$, localized response sampling $(n{h}^d{)}^{-1/2}$, and rectangular sampling $(nN{h}^d{\delta }^d{)}^{-1/2}$. The public tuning balances them. The approximation, positivity, and sampling ingredients are isolated in Lemmas 1, 2, and 3; their proofs are organized in Section A.

This theorem completes the main-body estimator construction. The approximation, sampling, and inversion arguments are organized in Section A.

Sharp precision and sample requirements

The sharp precision bound separates the information supplied by response labels from that supplied by all covariate-treatment records. The rate and attaining decision are defined once in Definitions 14 and 13. The following result gives their uniform minimax comparison, explicit sample-size regimes, and extension to randomized decisions.

The two terms express complementary precision requirements. The supplied-propensity term reflects the labeled-sample requirement for estimating the smooth contrast at a point; the product term reflects the joint requirement on labeled and treatment-record counts when the nuisance functions are unknown. If $S\ge S_{\mathrm{crit}}$, the labeled count already supports the supplied-propensity order. Equality $S=S_{\mathrm{crit}}$ belongs to this regime: then $q_{\ast }=1$, and the necessary inequality $N\ge n$ places every admissible sample pair at or beyond the saturation threshold. If $S<S_{\mathrm{crit}}$, the exponent $q_{\ast }>1$ creates an intermediate regime $n\le N\le n^{q_{\ast }}$, in which precision depends on the product $nN$. At $N=n^{q_{\ast }}$, the two terms coincide; for larger total counts, the supplied-propensity term governs the rate.

The lower bound concerns the original records and includes every Borel decision using the independent uniform randomizer. Thus the comparison applies to both deterministic and randomized estimation procedures. The original-record construction in Theorem 4 underlies the sparse-range nuisance converse in Lemma 6. That converse supplies the product-of-counts lower bound in the range where it governs precision. Combined with the supplied-propensity lower bound below, it yields the sharp comparison across all sample imbalances. The matching upper bound is attained by the publicly tuned decision through Theorem 1.

To interpret saturation, the next result evaluates the same primitive class when the designated propensity function is supplied to the decision. This benchmark identifies the labeled-sample precision against which the value of outcome-free records is measured. Its attaining known-design decision uses $$h=n^{-1/(2\gamma +d)},T^e({\mathcal{D}}_{n,m},U;e_P)={\mathrm{clip}}_{[-1,1]}\left[\frac{1}{n}\sum _{i=1}^n{w}_h(X_i)r_0^{\top }r(X_i)\left\{\frac{A_i{Y}_i}{e_P(X_i)}-\frac{(1-A_i)Y_i}{1-e_P(X_i)}\right\}\right].$$ Here $r_0=r(x_0)$, and $w_h$ and the degree-two basis $r$ use the centered localization measure from Definitions 7 and 8. These formulas are defined for every bandwidth $h>0$; the supplied-propensity smoother below uses only $h_n=n^{-1/(2\gamma +d)}$, which lies in $(0,1)$ for $n\ge 2$ and can exceed $1/2$ for small $n$. This is a known-design polynomial evaluation average, not an empirical local least-squares fit.