Introduction

This paper establishes matching finite-sample orders for private estimation and honest interval inference for a treatment effect at a fixed covariate value. The experiment consists of independent binary observational records with a scalar covariate, fixed treatment overlap, a bounded positive design density, rough nuisance regressions, and a Lipschitz treatment-effect contrast. Within this experiment, a single joint private release attains both the minimax expected absolute-error order and the optimal order of maximal expected length of connected intervals with uniform $90\%$ coverage. The matching conclusions concern the dependence on sample size and privacy budget, with explicit numerical constants.

Pointwise treatment effects describe how the expected response to treatment varies across observed characteristics. Under consistency and conditional exchangeability, observational treatment-arm regressions identify the conditional potential-outcome contrast. Pointwise evaluation additionally requires a choice of regression version. In the class studied here, contrast continuity and positive covariate density determine a unique continuous representative, making its value at the public evaluation location an identified feature of the observed-record law, as established in Lemma 1. Privacy then constrains how that feature can be estimated and reported from confidential records.

The statistical class in Definition 9 fixes the covariate domain to $[0,1]$, the evaluation location to $1/2$, and a common regularity radius of $3$. It combines a measurable density bounded above and away from zero, treatment overlap, one-tenth Hölder propensity and control regressions, and a Lipschitz contrast for binary outcomes. Releases satisfy pure central differential privacy under replacement adjacency on the full ordered-dataset space.

The smooth contrast permits the two treatment-arm means to share rough local variation that cancels in their difference, while rough treatment selection remains an additional nuisance. This scalar binary experiment makes the interaction between nuisance roughness, local occupancy, and privacy explicit.

Let $n$ denote sample size and $ϵ$ the privacy budget. For every integer $n\ge 2$ and $0<ϵ\le 1$, define the accuracy benchmark $$r(n,ϵ)=\min \left\{1,\max \left\{n^{-1/4},(n^2ϵ{)}^{-1/7},(nϵ{)}^{-1/2}\right\}\right\}.$$ Theorem 2 establishes that minimax maximal expected absolute error is bounded above and below by numerical multiples of this benchmark. Theorem 3 establishes the same order for minimax maximal expected length among private connected-interval procedures satisfying uniform finite-sample $90\%$ coverage. Both criteria integrate sampling variation and release randomization. The lower bounds range over all everywhere-defined private randomized procedures admitted by the respective criteria.

The attaining construction uses a public window around the evaluation location, partitioned into equal cells. Within each cell, treatment–outcome pair differences form a numerator and treatment pair differences form a denominator. Count weighting gives their population means the same occupancy weights, so the population ratio preserves the cellwise approximation to the target across every admissible measurable design density. The joint statistic has uniformly bounded replacement sensitivity, including changes involving cells with fewer than two records. Independent Laplace perturbations, a public denominator floor, and clipping therefore yield an everywhere-defined private ratio. Public tuning balances localization bias, within-cell nuisance approximation, pair variation, and privacy noise. The uniform squared-error envelope in Theorem 1 supplies an honest interval radius, and candidate-value inversion in Algorithm 3 reproduces that interval from the same joint release.

The converse uses causal alternatives satisfying the same model restrictions. Shared-sign alternatives vary the nuisance functions while maintaining a common treatment-effect separation; their mixture permits likelihood and component-coupling comparisons. Direct localized alternatives provide the complementary privacy comparisons. The finite testing priors in Lemma 7 combine these experiments across budget regimes. For scalar estimation, accurate recovery of separated targets would distinguish their released-output laws. For honest intervals, transferred coverage probabilities force simultaneous containment of separated targets with positive probability, and connectedness converts that containment into a length lower bound. This interval argument establishes the expected-length converse directly.

The closest pointwise comparison is Kennedy et al. (2024). At dimension one, contrast smoothness one, and average nuisance smoothness one tenth, their rate specializes to $n^{-1/4}$, with the nuisance average below the relevant one-sixth threshold. For the binary bounded-positive-density experiment here, shared occupancy weighting attains this statistical scale together with the privacy scales $(n^2ϵ{)}^{-1/7}$ and $(nϵ{)}^{-1/2}$. The reusable contribution is the combination of density-robust occupancy-weighted attainment with matching private testing comparisons; the connected-interval converse then transfers that frontier directly to honest expected length. Section 2 gives the detailed comparison of experiments, assumptions, and adjacent private and inferential literature.

The benchmark gives an order-based guide to sample-size and budget choices. As the budget decreases, its dominant term moves from statistical variation to sparse-pair privacy noise, then dense-occupancy privacy noise, and finally bounded-range saturation. Proposition 1 identifies the transition budgets and establishes that vanishing private minimax absolute risk along admissible budget sequences is equivalent to $nϵ\to \infty$. The interval comparison gives the same condition for vanishing optimal honest expected length.

The paper proceeds through related work, the observation model and decision criteria, and the within-cell private release. It then presents the matching estimation and honest interval frontiers, followed by discussion and limitations. The appendices develop identification and upper-bound ingredients, lower-bound experiments and component coupling, testing reductions, and proofs of the main results.

Related work

The closest comparison is with Kennedy et al. (2024), who characterize pointwise conditional average treatment effect estimation under separate Hölder restrictions on the propensity score, control regression, and treatment contrast. For their formula, dimension one, contrast smoothness one, and average nuisance smoothness one tenth give the rate $n^{-1/4}$; one tenth is below the corresponding one-sixth threshold. The present choice therefore isolates a simple rough-nuisance regime in which higher-order bias control is relevant. Their lower bound applies to a broader experiment allowing real outcomes; its construction uses a binary-outcome submodel and a covariate density bounded above that can vanish locally. Their higher-order local-polynomial estimator attains the stated rate under additional approximation, eigenvalue, boundedness, and covariate-distribution estimation conditions. Remark 11 expresses the last requirement through a third-order product involving design estimation and nuisance errors. The present experiment in Definition 9 combines binary outcomes, a scalar covariate, a density bounded above and away from zero, fixed overlap, one-tenth Hölder nuisance functions, and a Lipschitz treatment contrast. Within this class, Theorem 2 gives matching orders for private pointwise expected absolute error, with additional scales $(n^2ϵ{)}^{-1/7}$ and $(nϵ{)}^{-1/2}$. The comparison therefore concerns both the statistical experiment and the attainment argument: here, occupancy-weighted within-cell pairs support private attainment under the stated density bounds.

This construction connects to the literature on private U-statistics. Chaudhuri et al. (2024) give a pure differentially private estimator for the expectation of a bounded symmetric statistic, with a probability bound involving its variance, concentration of local Hájek projections, and privacy terms. The release in Algorithm 1 combines two count-weighted pair statistics within a public partition, adds Laplace noise jointly, and forms a clipped ratio with a public denominator floor. Its analysis addresses the random number of observations contributing pairs and the normalization needed for a local treatment contrast. The shared occupancy weights in the population numerator and denominator are central to Lemma 2, while Lemma 3 controls replacement sensitivity across cell counts, including transitions involving cells with fewer than two observations. Together, these features support the uniform guarantee in Theorem 1.

The lower-bound analysis draws on coupling-based private testing. Acharya et al. (2021) relate private distinguishability to the expected Hamming distance between datasets under a coupling, including comparisons between mixtures of product laws. The present analysis specializes this approach to causal alternatives satisfying the binary experiment and its smoothness restrictions. The common-mass construction in Definition 23 operates on conditional components of the shared-sign experiment, and Lemmas 5 and 6 connect these comparisons to private output laws. The separated testing priors in Lemma 7 then supply the experiments used for the estimation and interval lower bounds. The contribution lies in constructing and controlling these comparisons within the specified causal class.

Private heterogeneous-effect learners provide a complementary perspective. Niu et al. (2022) develop a general procedure for conditional average treatment effect learning under approximate differential privacy, using sample splitting and private base learners. Their implementations include single-stage and multistage estimators, and their empirical performance criterion is test-set mean squared error, with a bias–variance decomposition. Schröder et al. (2025) instead establish approximate differential privacy for a prespecified finite vector of conditional treatment effect queries through Gaussian output perturbation calibrated to the second-stage learner’s sensitivity. Their accompanying performance analysis concerns Neyman orthogonality and quasi-oracle estimation under nuisance estimation conditions (Schröder et al., 2025). The present estimand is the treatment contrast at a fixed covariate location, privacy is pure differential privacy under replacement adjacency as defined in Definition 10, and performance is measured by maximal expected absolute error and honest maximal expected interval length in Definitions 11 and 13. These choices make the comparisons specific to the estimand, privacy requirement, and decision criterion.

The methodological background includes residualization in semiparametric regression (Robinson, 1988), orthogonal scores and cross-fitting (Chernozhukov et al., 2018), and two-stage conditional treatment effect estimation (Nie et al., 2021; Kennedy, 2023). Higher-order influence functions supply a general framework for controlling nonlinear nuisance bias (Robins et al., 2008). Causal forests and generalized random forests provide adaptive local estimation and asymptotic inference under their respective regularity conditions (Wager et al., 2018; Athey et al., 2019). These approaches explain the importance of separating contrast regularity from nuisance complexity; the present pair construction gives a concrete private implementation for the rough-nuisance experiment in Definition 9.

Smooth-nuisance kernel results form a separate model comparison. Kim (2026) study conditional treatment effects under reproducing kernel Hilbert space restrictions on response functions, with additional spectral, source, or dimension structure for the contrast. Their kernel ridge regression analysis relates estimation rates to contrast complexity and overlap within those models. Here, regularity is specified directly through the Hölder and Lipschitz conditions in Definition 9, and the private risk comparison is established for that class. The distinction between these function spaces and performance criteria governs the interpretation of their respective optimality results.

Finally, the interval criterion connects private causal inference to honest nonparametric inference. Armstrong et al. (2018) derive finite-sample optimal intervals and efficiency bounds for linear regression functionals under Gaussian errors with known variance, while Armstrong et al. (2020) develop intervals whose critical values account explicitly for worst-case bias. The present construction likewise incorporates a public error envelope, and Theorem 3 establishes matching orders, up to constant factors, for maximal expected length subject to uniform finite-sample $90\%$ coverage over the causal class and pure replacement privacy. A complementary private causal inference result is provided by Schröder et al. (2025), who construct approximate differentially private intervals for the average treatment effect using asymptotic normality and accounting for uncertainty from estimation and privatization. The present expected-length result places pointwise inference for a Lipschitz contrast with rough nuisance functions within a uniform finite-sample decision framework.

Observation model, identification, and decision criteria

Generic constants $c$ and $C$ denote positive numerical constants whose values may change across occurrences. Absolute values denote scalar magnitudes, and $a\asymp b$ means that $a$ is bounded above and below by positive numerical multiples of $b$. For sample size $n$, record indices run from $1$ to $n$; sums over empty index sets equal zero. Expectations and probabilities with procedure subscripts integrate the procedure’s randomization as well as the observed sample.

Causal records and observational sampling

The full causal record is $(X,A,(Y(0),Y(1)),Y)$, where $X\in [0,1]$ is the scalar covariate, $A\in \{0,1\}$ is treatment, and the potential and observed outcomes are binary. Its observed marginal retains $(X,A,Y)$, and an ordered sample consists of $n$ independent records from that marginal. The anchored definitions immediately below record the coordinate, marginal-law, and dataset conventions used by the statistical class and decision problems.

The model uses a common public regularity radius $L$ for the nuisance functions and the treatment-effect contrast. Its numerical value is fixed throughout the analysis.

The causal record contains the covariate $X$, treatment indicator $A$, potential-outcome pair $Y^{\mathrm{pot}}$, and observed outcome $Y$. The following coordinate conventions distinguish the observed variables from the potential outcomes.

A causal law $P$ specifies the joint distribution of the full record. Its observational marginal $P^{\mathrm{obs}}$ specifies the distribution available to an estimator; the covariate marginal $P_X$ describes the random design.

Write $Y(A)=(1-A)Y(0)+AY(1)$ for the potential outcome selected by the realized treatment. The observed-record space $\mathcal{O}$ retains the covariate, treatment, and observed outcome, and an ordered dataset $D$ collects these records.

Sampling and causal identification rest on the following three conditions. The first specifies how observed records enter the dataset.

Independent and identically distributed observational sampling makes the common observed-record law the statistical experiment for estimation and inference (Kennedy et al., 2024). The next condition connects each observed outcome to the corresponding potential outcome.

Consistency assigns the realized outcome to the treatment actually received, giving the observed outcome regression its potential-outcome interpretation (Kennedy et al., 2024). Identification further requires that the covariate account for treatment selection in the following sense.

Conditional exchangeability permits comparison of the two treatment arms at a common covariate value (Kennedy et al., 2024). Together with consistency and the overlap restriction below, it identifies the conditional mean potential-outcome contrast from observational regressions, initially up to covariate-null sets.

Selected regressions and the statistical class

Pointwise estimation requires explicit regression versions. We therefore specify the selected design density $f_P$, propensity regression $e_P$, and control outcome regression ${\mu }_{0,P}$ before imposing their regularity conditions.

The selected treated outcome regression ${\mu }_{1,P}$ completes the observational contrast ${\tau }_P$. We use the selected conditional versions throughout and define $${\mu }_{1,P}(x)=E_P[Y\mid X=x,A=1],{\tau }_P(x)={\mu }_{1,P}(x)-{\mu }_{0,P}(x),x_0=\frac{1}{2},\theta (P)={\tau }_P(x_0).$$ Thus the fixed public evaluation location $x_0$ determines the scalar target $\theta (P)$.

Binary outcomes place both the contrast and the target in $[-1,1]$. We next organize the random-design, overlap, and smoothness restrictions that determine the statistical class.

The bounded positive random-design density ensures that covariate neighborhoods receive probability comparable to their Lebesgue length. These fixed numerical bounds instantiate the density condition in (Wager et al., 2018). Treatment probabilities are likewise bounded away from their endpoints.

Uniform treatment overlap ensures that both treatment arms remain represented throughout the covariate domain; the constants give a fixed numerical version of the standard overlap condition (Kennedy et al., 2024). The nuisance regressions may vary substantially over short distances, subject to the following common Hölder order.

Hölder propensity regularity controls local changes in treatment selection while allowing a rough propensity function of order $1/10$ (Kennedy et al., 2024). The control regression obeys the corresponding restriction.

Hölder control-regression regularity bounds local changes in the baseline outcome mean at the same rough order (Kennedy et al., 2024). The treatment-effect contrast has greater regularity than these nuisance functions.

Lipschitz contrast regularity controls the change in the conditional treatment effect over a covariate neighborhood. It is the Hölder order-one condition under the convention used in (Kennedy et al., 2024). Thus the model combines a Lipschitz target function with propensity and control regressions of Hölder order $1/10$, all within the fixed radius in Definition 1.

We collect these statistical restrictions into a class of causal laws. The private-release requirement is introduced afterward, once its output-space notation is in place.

The statistical class includes the identifying conditions and the design and function restrictions just stated.

For laws in Definition 9, consistency, exchangeability, and overlap identify the observational contrast with the conditional mean of $Y(1)-Y(0)$ almost everywhere under the covariate law. Contrast continuity and the positive design density then determine its pointwise interpretation: continuous versions agreeing almost everywhere agree throughout $[0,1]$. Consequently, the target $\theta (P)$ is the value of the continuous causal contrast at the fixed evaluation location. The argument is given in Lemma 1.

Private releases and decision criteria

Privacy constrains a release on the full ordered dataset space in Definition 2. Its comparison concerns an arbitrary dataset and a comparison dataset $D^{\prime }$, using replacement Hamming distance $d_{\mathrm{H}}(D,D^{\prime })$. A randomized release $M$ takes values in a standard Borel output space $\mathcal{B}$; the public privacy budget $ϵ$ bounds the change in its output probabilities.

The privacy requirement applies to every replacement-adjacent pair, including datasets outside any particular sampling law’s support.

Pure central differential privacy under replacement adjacency bounds the multiplicative change in every output-event probability when one observed record is replaced (Acharya et al., 2021). Requiring this property everywhere makes privacy a condition on the release itself, uniformly across the full dataset space. The admissible class is therefore defined independently of the causal law.

We use separate decision criteria for scalar estimation and interval inference. For a randomized scalar estimator $T$, the minimax absolute-error risk $R_{n,ϵ}$ measures the smallest achievable worst-case expected distance from the pointwise target.

The supremum evaluates accuracy uniformly over the causal class in Definition 9, while the infimum ranges over all private scalar releases in Definition 10. Interval inference instead evaluates expected length subject to a uniform coverage requirement. To express that criterion for randomized procedures, we first specify the measurable interval output space $\mathcal{I}$.

This coding accommodates all connected Borel intervals and makes coverage events and interval lengths measurable. For a randomized interval procedure $I$, the honest expected-length criterion $H_{n,ϵ}$ minimizes maximal expected length among private procedures attaining at least $90\%$ coverage for every law in the class.

Honesty here means uniform coverage over the entire causal class at the given sample size and privacy budget. Connectedness gives length its usual interpretation as the span of a confidence interval, and both coverage and expected length account for sampling and release randomization. Since the target lies in $[-1,1]$, the constant release of that interval is private, has coverage one, and supplies a finite-length admissible procedure. The two criteria in Definitions 11 and 13 thus evaluate estimation accuracy and honest interval precision within the same observational experiment and privacy model.

A private release based on within-cell pairs

A single release of local pair statistics supplies both an estimator for the pointwise treatment effect and an honest confidence interval under the criteria in Definitions 11 and 13. The construction uses public cell geometry and gives the numerator and denominator the same occupancy weights. This shared weighting accommodates every measurable design density satisfying Assumption 4. We first specify the clipping operation used to keep the reported estimate within the target range.

Localization uses a public radius $h$ around $x_0=1/2$. A public cell count $k$ determines the cell width $\delta$ and the equal-cell partition $(C_j{)}_{j=1}^k$. The cell occupancy $N_j(D)$ counts records with their dataset multiplicities. The following geometry fixes these quantities before the release is computed.

The public occupancy scale $\mathcal{A}$ measures the effective amount of information available from cells containing pairs: $$\mathcal{A}=nh\min \{1,n\delta \}.$$ The factor $nh$ describes the local sample size, while $\min \{1,n\delta \}$ accounts for the availability of a second record in the same cell. Thus the scale accommodates both sparse and densely occupied cells.

For the procedures below, define the public mean-squared-error envelope $V(h,\delta )$ and its conservative bias component $b_{\mathrm{env}}$ by $$V(h,\delta )=2^{20}\left(h^2+{\delta }^{2/5}+{\mathcal{A}}^{-1}+(ϵ\mathcal{A}{)}^{-2}\right),b_{\mathrm{env}}=2^{10}\sqrt{h^2+{\delta }^{2/5}}.$$ Here $b_{\mathrm{env}}^2$ collects the localization and within-cell bias terms in the envelope. The appendix separately writes $b_{\mathrm{cell}}=3h+24{\delta }^{1/5}$ for the sharper cellwise population certificate. The numerical benchmark $r(n,ϵ)$, which will determine the public tuning, is $$r(n,ϵ)=\min \left\{1,\max \left\{n^{-1/4},(n^2ϵ{)}^{-1/7},(nϵ{)}^{-1/2}\right\}\right\}.$$ All of these quantities depend on public parameters.

The numerator statistic $S_N(D)$ aggregates within-cell treatment–outcome differences, and the denominator statistic $S_D(D)$ aggregates within-cell treatment differences. Their common occupancy weighting explains the construction’s treatment of the design density. To describe that weighting, let $(X,A,Y)$ and $(X^{\prime },A^{\prime },Y^{\prime })$ be independent records drawn from the observed law conditional on membership in cell $C_j$. Define the conditional pair moments ${\nu }_j$ and $d_j$, and the occupancy weight $w_j$, by $${\nu }_j=E\{(A-A^{\prime })(Y-Y^{\prime })\mid X,X^{\prime }\in C_j\},d_j=E\{(A-A^{\prime }{)}^2\mid X,X^{\prime }\in C_j\},w_j=E\{N_j(D)1\{N_j(D)\ge 2\}\}.$$ The population means $\overline{N}$ and $\overline{D}$ of the two statistics consequently take the form $$\overline{N}=\sum _{j=1}^k{w}_j{\nu }_j,\overline{D}=\sum _{j=1}^k{w}_j{d}_j.$$ Both sums use the same weights, including the effect of cells containing fewer than two records. Their ratio therefore averages the cell ratios with weights proportional to $w_j{d}_j$. The smoothness restrictions in Assumptions 6, 7, and 8 control the cellwise approximation to the target, while the measurable density determines the common weighting. The detailed occupancy argument is given in Lemma 2.

The first procedure computes a single joint noisy release $\tilde{S}$. A positive public denominator floor $d_0$ then stabilizes the ratio. The local estimator ${\hat{T}}_{h,k}$ and local interval ${\hat{I}}_{h,k}$ are both postprocessings of that release.

The cell formulas are count-weighted sums of symmetric pair contributions. They can be computed from cell totals, preserving the pair construction while retaining a direct implementation. Cells with zero or one record contribute zero to both coordinates. Replacement privacy requires control of changes across all cell counts, including replacements that move a record between cells; Lemma 3 supplies this sensitivity argument. The independent Laplace perturbations implement the joint release, and the estimator and interval inherit its privacy by postprocessing (Dwork et al., 2006; Dwork et al., 2014).

Public tuning selects the estimator ${\hat{T}}^{\ast }$ and interval ${\hat{I}}^{\ast }$ using the benchmark alone. In the local branch, the dyadic choice of $h$ makes $k$ an integer and sets $\delta =h^5$, balancing the two squared-bias terms. The following procedure also specifies the constant outputs for the branch $r(n,ϵ)\ge 1/8$.

The stated interval is deliberately conservative at moderate sample sizes. In the local branch, $\delta =h^5$, so $V(h,\delta )\ge 2^{21}{h}^2$. The dyadic rule gives $h>r(n,ϵ)/2$, while $r(n,ϵ)\ge n^{-1/4}$; hence $$10V(h,\delta )>5\cdot 2^{20}{n}^{-1/2}.$$ For $2\le n\le 2^{40}$, its square root exceeds the diameter of $[-1,1]$, and the clipped centered interval is therefore exactly $[-1,1]$, with the theorem’s uniform coverage guarantee. The fallback branch has the same output. The practical interpretation of these constants is discussed under Limitations and future work.

The interval also has a candidate-value interpretation. For each candidate effect $\vartheta$, inversion compares a clipped numerator with the candidate multiplied by the stabilized denominator. The resulting interval $I_{n,ϵ}^{\ast }$ uses the same joint release as the point estimate. To express the connected output of the acceptance rule, we first define the interval hull.

With this operation, the inversion procedure expresses the interval directly in terms of the released coordinates and public error envelope. In the inversion algorithm, the symbol $b$ denotes $b_{\mathrm{env}}$.

Because $B_D$ is positive, dividing the acceptance inequality by $B_D$ gives an interval centered at $B_N/B_D={\hat{T}}_{h,k}$, with radius $\rho$, intersected with $[-1,1]$. The acceptance set contains its center and is connected. Hence the inversion reproduces the interval in Algorithms 1 and 2. Computing the point estimate and this interval together uses one draw of the joint noise.

The next result establishes privacy for every dataset and uniform statistical guarantees over the causal class. It gives both the local squared-error envelope and the absolute-error and expected-length bounds obtained by public tuning.

The four terms in the error envelope separate the statistical mechanisms. Localization of the Lipschitz contrast contributes $h^2$, and within-cell approximation of the two rough nuisance functions contributes ${\delta }^{2/5}$. Pair-statistic variation contributes at scale ${\mathcal{A}}^{-1}$. The replacement-copy variance inequality of Boucheron et al. (2013) controls this variation through changes induced by replacing individual independent records. Denominator stabilization keeps the noisy ratio defined across all occupancies, while the Laplace second moments contribute the term $(ϵ\mathcal{A}{)}^{-2}$. The occupancy and replacement-sensitivity details appear in Lemmas 2 and 3.

Under the public choice $\delta =h^5$, the two squared-bias terms have the same order. The benchmark balances this bias against pair variation and privacy noise in the sparse and dense occupancy regimes. Finally, the squared-error bound supplies the $90\%$ coverage guarantee through the radius $\sqrt{10V(h,\delta )}$. Intersecting the interval with the target range preserves coverage and bounds its length by two. Thus Theorem 1 supplies a common private release with uniform estimation and finite-sample interval guarantees throughout the causal class in Definition 9.

Optimal estimation and honest confidence intervals

The uniform guarantees in Theorem 1 lead to two decision-theoretic conclusions. The first characterizes the smallest maximal expected absolute error among all pure-private scalar procedures. The second characterizes the smallest maximal expected length among pure-private connected intervals with class-wide $90\%$ coverage. Both criteria are governed by the same numerical benchmark, under the causal-law restrictions in Definition 9.

For scalar estimation, the comparison includes every everywhere-defined private randomized procedure admitted by Definition 10. The publicly tuned estimator of Algorithm 2 attains the optimal order uniformly over the complete model class, including its admissible measurable covariate densities.

The benchmark expresses three competing accuracy scales. The term $n^{-1/4}$ reflects the statistical balance between localization bias and variation of the within-cell pairs. The two budget-dependent terms reflect the interaction of privacy noise with sparse and dense cell occupancy. Public tuning balances these contributions, and the outer truncation accommodates the bounded target range. The lower bound establishes that their maximum is an unavoidable order of maximal absolute risk over the stated class. The constants are explicit finite-sample bounds; optimality here concerns the common order in sample size and privacy budget.

The lower-bound intuition is observational indistinguishability. Separated treatment-effect targets can be embedded in causal laws whose observed samples are sufficiently difficult to distinguish. Any scalar release that estimates both targets accurately would also distinguish the associated experiments. Privacy strengthens this comparison by controlling how much the distribution of a released statistic can change when records are replaced. The testing priors and the private release comparison are presented in Lemmas 7 and 6; together they connect target separation to a lower bound that applies to every admissible scalar procedure.

The construction of the sign-indexed alternatives and their component couplings appears in Definition 19, Definition 23, and Lemma 5, while Definition 24 supplies the direct comparison. These constructions organize the observational and privacy comparisons within the same causal model class. The quantitative arguments use the replacement-copy variance inequality of Boucheron et al. (2013), the bounded-differences exponential-moment bound in Boucheron et al. (2013), and the labelled-tree enumeration formula of Keller et al. (2017). The appendix places these tools alongside the experiments to which they apply.

Honest inference imposes a distinct decision criterion: coverage must hold for every law in the class, while expected length is assessed in the worst case, as specified in Definition 13. Define the interval benchmark $\ell (n,ϵ)$ by $$\ell (n,ϵ)=r(n,ϵ),$$ and set the lower and upper numerical constants $c_I$ and $C_I$ to $$c_I=2^{-20},C_I=2^{22}.$$ With these conventions, the candidate-value inversion of Algorithm 3 attains the same order for honest expected length.

The theorem below has two branches: the full-range interval applies when $r(n,ϵ)\ge 1/8$, while every subsequent display involving $h$, $k$, $\delta$, $\mathcal{A}$, $V$, and the centered pair-ratio interval applies only on the local branch $r(n,ϵ)<1/8$.

The upper bound turns the uniform mean-squared-error envelope into a public interval radius. The factor $\sqrt{10}$ yields the stated coverage level, and intersection with the target range preserves containment of the true effect. Both endpoints are functions of the single joint private release and public quantities. Thus the estimator and interval share a release, while the interval guarantee controls coverage separately at every member of the causal class. The expected-length bound then places this particular procedure within a fixed numerical factor of the optimal honest length.

The interval converse has its own geometric intuition. Honesty requires containment of each target under its corresponding experiment. Observational indistinguishability and privacy contraction transfer these containment probabilities to a common output distribution. With positive probability, the released interval must consequently contain both separated targets. Connectedness forces its length on that event to be at least their separation. This simultaneous-containment argument links the experiments in Lemma 7 directly to the expected-length criterion. It explains why honest connected intervals inherit the same numerical order as scalar estimation, with the explicit comparison between the two criteria given in Theorem 3.

The benchmark also provides a direct guide to sample-size and budget choices. Its transition points identify which accuracy scale is largest, and its behavior along budget sequences determines uniform consistency.

For each fixed sample size, Proposition 1 gives the following budget regimes. The expressions agree at every shared endpoint. $$\begin{array}{lll}\text{Privacy budget}& \text{Benchmark}& \text{Dominant scale}\\ n^{-1/4}\le ϵ\le 1& n^{-1/4}& \text{Statistical variation}\\ n^{-3/5}\le ϵ\le n^{-1/4}& (n^2ϵ{)}^{-1/7}& \text{Sparse-pair privacy noise}\\ n^{-1}\le ϵ\le n^{-3/5}& (nϵ{)}^{-1/2}& \text{Dense-occupancy privacy noise}\\ 0<ϵ\le n^{-1}& 1& \text{Bounded-range saturation}\end{array}$$

The first regime includes the budget $ϵ=1$, at which the benchmark is $n^{-1/4}$. As the budget decreases, the two privacy terms successively determine the optimal order. In the saturation regime, the lower bound in Theorem 2 keeps maximal absolute risk bounded away from zero, and the interval lower bound in Theorem 3 gives the corresponding positive expected-length bound.

For a sequence of admissible budgets, the decisive consistency condition is $n{ϵ}_n\to +\infty$. By Proposition 1, this condition is necessary and sufficient for the private minimax absolute risk to vanish. The comparison in Theorem 3 yields the same criterion for vanishing optimal honest expected length. Consequently, the two principal results characterize both uniform estimation accuracy and the attainable length of honest connected intervals through a common sample-size and privacy-budget scale.

Discussion and limitations

For the binary observational class in Definition 9, Theorems 2 and 3 give the common order $r(n,ϵ)$ for private point estimation and honest expected interval length. The lower bounds range over every procedure admitted by the respective decision criteria, while the pair-ratio release supplies attainment.

The shared occupancy weighting is the main reusable feature of the upper-bound construction. It gives the population numerator and denominator common density-dependent occupancy weights, while the occupancy scale $\mathcal{A}=nh\min \{1,n\delta \}$ measures the availability of usable pairs in localized cells. Nuisance roughness determines the required cell resolution, and privacy noise is then amplified by the same occupancy shortage. This construction shows how density robustness, rough nuisances, and privacy interact through the occupancy scale.

The two private scales have different origins. The term $(n^2ϵ{)}^{-1/7}$ reflects privacy in the sparse-pair regime, while $(nϵ{)}^{-1/2}$ reflects privacy once cells are densely occupied. Their transition to saturation yields the consistency condition $n{ϵ}_n\to +\infty$, and the regime comparison characterizes optimal dependence on sample size and privacy budget within the specified model.

Honest uncertainty quantification shares the estimation benchmark through complementary upper and lower arguments. The upper bound converts the uniform squared-error envelope into an interval radius. For the converse, transferred coverage forces a connected interval to contain two separated targets with positive probability under a common release distribution, and connectedness converts that separation into expected length. This simultaneous-containment argument establishes the interval frontier directly. Privacy is uniform over adjacent datasets, while error and coverage are uniform over laws in the causal class and include release randomization.

Limitations and future work

The certified interval constants are extremely conservative. As shown after Algorithm 2, the local-branch radius already exceeds two for every $2\le n\le 2^{40}$, so the reported interval is deterministically $[-1,1]$ throughout that range; the fallback branch is also full-range. The interval frontier should therefore be read as an asymptotic order result and a finite-sample coverage certificate, rather than as a claim of useful finite-sample width at these constants.

The regime identities characterize rates and do not by themselves prescribe finite-sample sample-size or privacy-budget choices.

The release is specified using real-valued arithmetic and ideal Laplace random variables. A represented-real implementation requires explicit treatment of numerical rounding, noise generation, denominator stabilization, and interval endpoints. Software privacy certification is a separate task: it must establish the privacy guarantee for the implemented output distribution and account for the numerical operations used by the program.

The minimax comparisons range over pure-private procedures with budgets in $(0,1]$. Establishing the unrestricted nonprivate minimax risk is a separate comparison. In particular, the benchmark at $ϵ=1$ describes the stated private decision problem; identifying its relation to an unrestricted nonprivate optimum requires an appropriate converse and matching construction for that larger procedure class.

The model uses binary treatment and outcomes, a scalar covariate, independent records, and fixed density, overlap, and smoothness restrictions. Broader outcome distributions, higher-dimensional covariates, and dependent or alternative sampling designs call for corresponding identification conditions, sensitivity bounds, and lower-bound experiments. Allowing overlap to vary with sample size or changing the nuisance and contrast smoothness classes would likewise require a new analysis of the bias, occupancy, and privacy balance. Adaptation to unknown smoothness introduces the additional question of selecting resolution while preserving privacy and honest coverage.

Finally, the results evaluate worst-case decision criteria and supply a concrete procedure attaining their optimal orders. Evaluation of particular causal-effect estimators remains a useful direction, including comparisons of finite-sample error, coverage, interval length, and computational cost under specified data-generating laws. Such evaluations would clarify how the explicit construction and its conservative error envelope perform at sample sizes and privacy budgets relevant to empirical applications.

Appendices

Identification and upper-bound proofs

The auxiliary results establish the pointwise interpretation of the target, the population bias of the pair ratio, and the stability of its two coordinates. Their statistical roles are complementary: identification fixes the estimand, common occupancy weights preserve the cellwise approximation, and replacement bounds support both privacy and variance control. Together they supply the ingredients used in Theorem 1.

Point identification

The selected contrast satisfies the Lipschitz restriction in Assumption 8. Its continuity, together with the positive density in Assumption 4, determines a unique continuous version throughout the covariate domain. The following result connects this version to the potential-outcome contrast and establishes that its value at the evaluation point is determined by the observed-record law.