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A Minimax Lower Bound for Interior Dose-Response Estimation

Abstract

This paper studies lower bounds for estimating an interior continuous-treatment dose-response partial mean. The target is $θ_{P} (t_{0}) = \int μ_{P} (t_{0}, x) p_{X, P} (x) d x,$ interpretable as a causal dose-response mean under consistency, no unmeasured confounding, and local positivity. Over the same Hölder dose class $P_{α, β, s} (M, c_{0}, ε_{0}, t_{0})$ , under iid sampling, bounded outcomes, an interior evaluation dose, the stated anisotropic Hölder restrictions, and a strict-slack baseline condition, we prove the same-class minimax lower bound $R_{n} {P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}} \geq c n^{- 2 α / (2 α + 1)}$ for all sufficiently large $n$ . The lower-bound exponent is the one-dimensional treatment-regression exponent and holds for every fixed $β > 0$ , although the constant and slack-baseline feasibility may depend on $β$ . We compare this lower floor with the published higher-order influence-function benchmark $ρ_{n} = n^{- 2 α / (2 α + 1)} \lor n^{- 2/ (1 + d / (4 s) + 1/ α)} .$ When $d \leq 4 s$ , the lower-bound exponent equals the exponent of this published comparator, whose upper theorem concerns a distinct localized-regularity class. When $4 s < d$ , the benchmark is governed by the covariate-smoothness term and has a strictly smaller exponent than the same-class lower-floor exponent. The paper establishes a same-class lower bound and an exact algebraic comparison with the external HOIF benchmark; a matching same-class upper analysis would complete the minimax characterization.

Introduction

Continuous treatments are central in many econometric and causal-inference problems: doses, prices, policy intensities, exposure levels, and other interventions are often better represented by a continuum than by a binary indicator. In the potential-outcome framework of Rubin (1974) and Robins (1986), the object of interest is commonly a dose-specific mean $E {Y (t)}$ . Under consistency, conditional ignorability, and positivity, this causal object is identified by an observed-data partial mean, averaging the conditional regression of $Y$ on $(A, X)$ at dose $t$ over the marginal distribution of covariates. This paper studies the difficulty of estimating that fixed-dose partial mean at an interior dose.

The formal target is Definition 1, $θ_{P} (t_{0}) = \int_{[0, 1]^{d}} μ_{P} (t_{0}, x) p_{X, P} (x) d x,$ where $A \in [0, 1]$ , $X \in [0, 1]^{d}$ , and $μ_{P} (a, x) = E_{P} [Y ∣ A = a, X = x]$ . The model class in Definition 2 imposes iid sampling, bounded outcomes, an interior evaluation dose, local positivity, Hölder smoothness of the outcome regression and treatment density in the treatment coordinate, and Hölder smoothness of the relevant covariate functions. The causal assumptions supply the interpretation of the same observed-data functional as a dose-response mean; the lower bound itself is a statement about the observed-data experiment and the minimax risk in Definition 3.

The continuous-treatment literature provides both the identification background and the estimator benchmarks for this problem. Early formulations include the generalized propensity-score and propensity-function approaches of Imbens (2000), Hirano et al. (2004), and Imai et al. (2004), with comparisons developed by Zhao et al. (2013). Semiparametric efficiency and influence-function methods provide the language for regular estimation and orthogonalization (Bickel et al., 1993; van der Vaart, 1998; van der Laan et al., 2003; Chernozhukov et al., 2018). For nonparametric dose-response curves, Kennedy et al. (2017), Lee (2018), and Colangelo et al. (2020) are direct antecedents. The closest upper-side benchmark for this paper is Bonvini et al. (2022), which builds on higher-order influence-function methods (Robins et al., 2008; Robins et al., 2015) and reports the rate recorded here as Definition 4.

The main result is a same-class lower bound. Under the hypotheses of Theorem 1, including positive smoothness $α, β, s$ , the stated interior and boundedness constants, and the strict-slack baseline condition in Assumption 12, there is a constant $c > 0$ such that, for all sufficiently large $n$ , $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) .$ This theorem establishes the lower half of the minimax analysis for the original Hölder dose class and fixes the treatment-regression obstruction at exponent $2 α / (2 α + 1)$ .

The comparison sequence is $ρ_{n} = n^{- 2 α / (2 α + 1)} \lor n^{- 2/ (1 + d / (4 s) + 1/ α)} .$ In the smooth-covariate regime $d \leq 4 s$ , Proposition 1 and Theorem 2 show that the lower-bound exponent equals the exponent of $ρ_{n}$ , giving an exact exponent comparison with the published benchmark. In the deficient-covariate regime $4 s < d$ , Theorem 3 shows that $ρ_{n}$ is governed by the covariate-smoothness expression and has a strictly smaller exponent than the same-class lower floor. This ordering isolates the same-class upper behavior as the remaining question in that regime.

The lower-bound construction can be viewed as embedding the classical one-dimensional pointwise nonparametric regression obstruction inside the continuous-treatment partial-mean experiment. The embedding must preserve the observed-data restrictions of Definition 2: bounded outcomes, local positivity around $t_{0}$ , smooth covariate density, treatment-density smoothness, and the causal dose-response interpretation. The strict-slack baseline condition in Assumption 12 is used to keep the least favorable alternatives inside the class while perturbing only the treatment regression. This is why the resulting lower exponent is insensitive to the fixed treatment-density smoothness index $β$ at the exponent level, while constants and feasibility may still depend on $β$ .

The scope is deliberately narrower than neighboring continuous-exposure problems. Recent work studies modified treatment policies, weak-overlap or alternative identification regimes, nonparametric inference for dose-response curves, and incremental continuous-exposure interventions (Hejazi et al., 2022; Doss et al., 2022; Hudson et al., 2023; Shi et al., 2024; Zhang et al., 2024; Schindl et al., 2024; Bonvini et al., 2024). Those estimands or positivity structures can change the role of the treatment density and the smoothing built into the target. The present paper concerns the fixed interior partial mean in Definition 1 under the same-class Hölder restrictions in Definition 2.

The precise proof scope is summarized in the verification appendix. The mathematical contribution is the same-class lower bound, the two algebraic comparisons with the published HOIF benchmark, and the explicit separation between those lower-bound statements and upper results proved for other localized-regularity classes.

Setup and observed-data class

We begin with the observed-data experiment. One observation is the observed unit $O = (Y, A, X)$ , where the outcome $Y$ is real-valued, the treatment dose $A$ lies in $[0, 1]$ , and the covariate vector $X$ lies in $[0, 1]^{d}$ . The single-observation law is denoted by $P$ ; its covariate marginal is $P_{X}$ , with density $p_{X, P}$ on $[0, 1]^{d}$ . For $a \in [0, 1]$ and $x \in [0, 1]^{d}$ , let the conditional mean and treatment density be $μ_{P} (a, x) = E_{P} [Y ∣ A = a, X = x], π_{P} (a ∣ x) = density of A ∣ X = x .$ The outcome regression $μ_{P}$ is the nuisance function whose value at the evaluation dose enters the target, while the conditional treatment density $π_{P}$ governs the local amount of information available near that dose.

The sample consists of $O_{1}$ , $\dots$ , $O_{n}$ , generated from the product law $P^{n}$ under the following sampling condition.

Assumption 1 [ass:iid-sampling] (Iid sampling).

The observations $O_{1}, \dots, O_{n}$ are iid draws from $P$ .

⊢ Lean

Assumption 1 is the standard iid sampling condition for the nonparametric experiment and fixes $P^{n}$ as the sampling law used in the risk calculations (Tsybakov, 2009).

Fix positive smoothness orders $α$ , $β$ , and $s$ , a common envelope and Hölder radius $M$ , a local positivity constant $c_{0}$ , an interior evaluation dose $t_{0}$ , and an interior half-width $ε_{0}$ . Throughout, $H^{γ} (M; D)$ denotes the Hölder ball of order $γ$ and radius $M$ on the domain $D$ . The analysis is local in the treatment coordinate, over the $[t_{0} - ε_{0}, t_{0} + ε_{0}]$ .

We impose the basic observed-data restrictions before introducing the causal interpretation.

Assumption 2 [ass:bounded-outcome] (Bounded outcome).

Under $P$ , $∣ Y ∣ \leq M$ almost surely.

⊢ Lean

Assumption 2 is the standard uniform bounded outcome condition; it keeps the loss and testing reductions within a common envelope over the model class (Bonvini et al., 2022).

Assumption 3 [ass:interior-dose] (Interior dose).

The exposure neighborhood around $t_{0}$ is contained in the interior of the exposure support: $[t_{0} - ε_{0}, t_{0} + ε_{0}] \subset (0, 1) .$

⊢ Lean

Assumption 3 is the standard interior evaluation point condition, excluding boundary effects in the continuous-treatment coordinate (Colangelo et al., 2020).

Assumption 4 [ass:local-positivity] (Local positivity).

For every $a \in [t_{0} - ε_{0}, t_{0} + ε_{0}]$ and every $x \in [0, 1]^{d}$ , $π_{P} (a ∣ x) \geq c_{0} .$

⊢ Lean

Assumption 4 is the standard local positivity condition; it requires enough treatment density in the neighborhood of $t_{0}$ for the interior dose-response target to be statistically meaningful (Bonvini et al., 2022).

The smoothness restrictions separate regularity in the treatment coordinate from regularity in the covariate coordinate. This separation is useful because the lower bound varies with $α$ , $β$ , $s$ , and $d$ through distinct approximation and testing constraints.

Assumption 5 [ass:mu-treatment-holder] (Treatment-regression dose smoothness).

For every $x \in [0, 1]^{d}$ , the map $a ⟼ μ_{P} (a, x)$ belongs to the Hölder ball $H^{α} (M; [t_{0} - ε_{0}, t_{0} + ε_{0}])$ .

⊢ Lean

Assumption 5 is the standard Hölder smoothness in the treatment coordinate for the regression function near the target dose (Tsybakov, 2009).

Assumption 6 [ass:pi-treatment-holder] (Treatment-density dose smoothness).

For every $x \in [0, 1]^{d}$ , the map $a ⟼ π_{P} (a ∣ x)$ belongs to the Hölder ball $H^{β} (M; [t_{0} - ε_{0}, t_{0} + ε_{0}])$ .

⊢ Lean

Assumption 6 is the standard Hölder smoothness condition for the conditional treatment density in the treatment coordinate (Bonvini et al., 2022).

Assumption 7 [ass:mu-covariate-holder] (Treatment-regression covariate smoothness).

The map $x ⟼ μ_{P} (t_{0}, x)$ belongs to the Hölder ball $H^{s} (M; [0, 1]^{d})$ .

⊢ Lean

Assumption 7 is the standard Hölder smoothness condition in the covariate coordinate for the target-dose regression surface (Tsybakov, 2009).

Assumption 8 [ass:pi-covariate-holder] (Treatment-density covariate smoothness).

The map $x ⟼ π_{P} (t_{0} ∣ x)$ belongs to the Hölder ball $H^{s} (M; [0, 1]^{d})$ .

⊢ Lean

Assumption 8 imposes the standard covariate smoothness condition on the marginal covariate density (Bonvini et al., 2022).

Assumption 9 [ass:px-holder] (Covariate-density smoothness).

The marginal covariate density $p_{X, P}$ belongs to the Hölder ball $H^{s} (M; [0, 1]^{d})$ and satisfies $0 \leq p_{X, P} (x) \leq M$ for every $x \in [0, 1]^{d}$ .

⊢ Lean

Assumption 9 is the standard Hölder-smooth covariate density condition, together with a uniform upper bound on the marginal density (Bonvini et al., 2022).

The statistical target is the partial mean of the target-dose regression surface over the covariate distribution.

Definition 1 [def:theta-functional] (Dose-response partial mean

θ_{P} (t_{0})

The dose-response partial mean at the evaluation dose $t_{0}$ is $θ_{P} (t_{0}) = \int_{[0, 1]^{d}} μ_{P} (t_{0}, x) p_{X, P} (x) d x .$

⊢ Lean

Definition 1 is an observed-data functional: it depends only on the conditional mean of $Y$ given $(A, X)$ and on the marginal law of $X$ . The next subsection overlays the causal interpretation under which this same functional is the dose-response mean.

The causal overlay introduces potential outcomes $Y (a)$ , indexed by $a \in [0, 1]$ , on the same probability space as the observed data. The following two assumptions are the identification conditions used to interpret Definition 1 causally.

Assumption 10 [ass:consistency] (Consistency).

Under $P$ , $Y = Y (A)$ almost surely.

⊢ Lean

Assumption 10 is the standard consistency condition, linking the observed outcome to the potential outcome at the realized dose (Kennedy et al., 2017).

Assumption 11 [ass:no-unmeasured-confounding] (No unmeasured confounding).

For every $a \in [0, 1]$ , the potential outcome $Y (a)$ is independent of $A$ conditional on $X$ under $P$ .

⊢ Lean

Assumption 11 is the standard conditional ignorability condition for continuous treatments (Kennedy et al., 2017). Together with local positivity in Assumption 4, it identifies the causal dose-response mean at the interior dose $t_{0}$ with the observed-data partial mean in Definition 1.

Combining these restrictions gives the observed-data class used for the lower-bound analysis. The minimax risk and the published benchmark rate referenced in the definition are introduced formally in Definition 3 and Definition 4.

Definition 2 [def:holder-dose-class] (Hölder Dose Class

P_{α, β, s} (M, c_{0}, ε_{0}, t_{0})

With the minimax risk $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0})$ and the published HOIF benchmark $ρ_{n}$ as in Definitions 3 and 4, define $P_{α, β, s} (M, c_{0}, ε_{0}, t_{0})$ to be the collection of observed-data laws $P$ satisfying all of the following conditions:

(Sampling.) The observations satisfy Assumption 1.
(Causal identification.) Consistency and no unmeasured confounding hold as in Assumptions 10 and 11.
(Outcome boundedness.) The bounded-outcome condition holds as in Assumption 2.
(Interior dose.) The evaluation dose $t_{0}$ and local radius $ε_{0}$ satisfy Assumption 3.
(Local positivity.) The treatment density obeys the local positivity condition with constant $c_{0}$ as in Assumption 4.
(Treatment-direction smoothness.) The maps $a \mapsto μ_{P} (a, x)$ and $a \mapsto π_{P} (a ∣ x)$ satisfy the treatment-direction Hölder conditions in Assumptions 5 and 6.
(Covariate-direction smoothness and density range.) The maps $x \mapsto μ_{P} (t_{0}, x)$ and $x \mapsto π_{P} (t_{0} ∣ x)$ satisfy the covariate-direction Hölder conditions in Assumptions 7 and 8, and $p_{X, P}$ satisfies the Hölder, nonnegativity, and upper-envelope condition in Assumption 9.
(Semantic ties to the observed-data law.) The regression, covariate-density, and treatment-density symbols are tied to $P$ : $μ_{P} (A, X)$ equals the conditional mean of $Y$ given $(A, X)$ $P$ -almost surely; $p_{X, P}$ is the density of the $X$ -marginal on $[0, 1]^{d}$ ; and $π_{P}$ is nonnegative on $[0, 1] \times [0, 1]^{d}$ with the $(A, X)$ -law density $(a, x) \mapsto π_{P} (a ∣ x) p_{X, P} (x)$ relative to Lebesgue measure on $[0, 1] \times [0, 1]^{d}$ .

⊢ Lean

Definition 2 is a statistical model for observed laws equipped with the stated causal interpretation. The lower-bound alternatives are constructed as observed-data laws in this class; the potential-outcome overlay supplies the causal reading of the same target while preserving the statistical experiment.

For the least favorable submodels used later, we also require a baseline law at a positive distance from the model boundary.

Assumption 12 [ass:baseline-submodel-slack] (Baseline submodel slack).

There exist a covariate density $p_{0}$ on $[0, 1]^{d}$ , a conditional treatment density $q_{0}$ on $[0, 1]$ , a constant $η_{0} > 0$ , and a constant $B_{0}$ with $0 < B_{0} < M$ , such that $p_{0} (x) \geq 0$ for all $x \in [0, 1]^{d}$ , $q_{0} (a) \geq 0$ for all $a \in R$ , $\int_{[0, 1]^{d}} p_{0} (x) d x = 1 and \int_{0}^{1} q_{0} (a) d a = 1,$ $∥ p_{0} ∥_{H^{s} ([0, 1]^{d})} \leq M - η_{0},$ $q_{0}$ has Hölder norm at most $M - η_{0}$ of order $β$ on $[t_{0} - ε_{0}, t_{0} + ε_{0}]$ , $q_{0} (a) \geq c_{0} + η_{0} for all a \in [t_{0} - ε_{0}, t_{0} + ε_{0}],$ and $p_{0} (x) \leq M - η_{0} for all x \in [0, 1]^{d} .$

⊢ Lean

Assumption 12 is specific to this analysis. It provides an interior baseline density $p_{0}$ , an interior baseline treatment density $q_{0}$ , and a positive margin $η_{0}$ , so that the perturbations used in the lower-bound construction can remain inside Definition 2.

Main results

The target in Definition 1 is evaluated under squared loss over the observed-data class in Definition 2. We first fix the risk criterion. The estimator is allowed to be any measurable function of the iid sample, with range restricted to the outcome envelope; this normalization is immaterial for the rate comparisons below because the target itself lies in the same bounded range.

Definition 3 [def:minimax-risk] (Minimax risk

R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0})

The minimax risk over the model class $P_{α, β, s} (M, c_{0}, ε_{0}, t_{0})$ at the evaluation dose $t_{0}$ is $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) = \hat{θ}_{n} measurable \hat{θ}_{n} (O_{1}, \dots, O_{n}) \in [- M, M] for every sample in f P \in P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}) sup E_{P^{\otimes n}} [{\hat{θ}_{n} (O_{1}, \dots, O_{n}) - θ_{P} (t_{0})}^{2}],$ where $P^{\otimes n}$ is the product law of the iid sample $O_{1}, \dots, O_{n}$ .

⊢ Lean

The benchmark used for comparison is the rate reported for higher-order influence-function methods in the continuous-exposure literature. We use $x \lor y$ to denote $max {x, y}$ .

Definition 4 [def:published-hoif-rate] (Published HOIF benchmark

ρ_{n}

The published higher-order influence-function benchmark rate is $ρ_{n} = n^{- 2 α / (2 α + 1)} \lor n^{- 2/ (1 + d / (4 s) + 1/ α)} .$

⊢ Lean

Definition 4 records the comparison sequence from the published localized-regularity upper result. The first term is the one-dimensional nonparametric regression scale in the treatment coordinate, while the second is the higher-order influence-function benchmark involving the covariate dimension and covariate smoothness (Robins et al., 2008; Robins et al., 2015; Kennedy et al., 2017; Bonvini et al., 2022). The lower bound below is the paper’s same-class result.

Theorem 1 [thm:sharp-pointwise-lower-bound] (Sharp Pointwise Lower Bound).

For every covariate dimension $d$ and every real constants $α, β, s, M, c_{0}, ε_{0}, t_{0}$ , suppose that

(Positive smoothness.) $α > 0$ , $β > 0$ , and $s > 0$ .
(Regime constants.) $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , and the local window determined by $(t_{0}, ε_{0})$ is contained in $(0, 1)$ .
(Strict-slack baseline.) Assumption 12 holds for $(d, β, s, M, c_{0}, ε_{0}, t_{0})$ .

Then there exists a constant $c > 0$ such that, for all sufficiently large sample sizes $n$ , $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}),$ where $P_{α, β, s} (M, c_{0}, ε_{0}, t_{0})$ is the model class of Definition 2 and $R_{n}$ is the minimax risk of Definition 3.

⊢ Lean

Theorem 1 gives a lower floor for the original Hölder dose class, uniformly over every positive value of the treatment-density smoothness parameter $β$ . Its construction varies the treatment regression while keeping the density components within the slack allowed by Assumption 12. The result fixes the same-class treatment-regression obstruction and supplies the lower half of the minimax analysis.

The next statements reduce the comparison with $ρ_{n}$ to algebraic regimes. When the covariate smoothness is high enough relative to dimension, the maximum in Definition 4 is attained by the treatment-regression term.

Proposition 1 [prop:oracle-regime-reduction] (Oracle Regime Reduction).

Let $d$ be the covariate dimension, and let $α, β, s, M, c_{0}, ε_{0}, t_{0}$ be constants. Suppose that

(Well-formed constants.) $α > 0$ , $β > 0$ , $s > 0$ , $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , and the window $[t_{0} - ε_{0}, t_{0} + ε_{0}]$ is contained in $(0, 1)$ .
(Oracle regime.) $d \leq 4 s$ .
(Baseline slack.) Assumption 12 holds for $d, β, s, M, c_{0}, ε_{0}, t_{0}$ .

Then there exists a constant $c > 0$ such that, for all sufficiently large sample sizes $n$ , $ρ_{n} = n^{- 2 α / (2 α + 1)}$ for the published HOIF benchmark rate $ρ_{n}$ of Definition 4, and $c ρ_{n} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}),$ where the model class is Definition 2 and the minimax risk is Definition 3.

⊢ Lean

Proposition 1 gives the exact comparison statement for $d \leq 4 s$ : the certified lower-bound exponent agrees with the exponent of the published benchmark sequence. Construction of an estimator over the same Hölder class is the corresponding upper-analysis question.

Theorem 2 [thm:sharp-minimax-smooth-covariate] (Smooth Covariate Minimax Floor).

Let $d \in N$ and let $α, β, s, M, c_{0}, ε_{0}, t_{0} \in R$ . Suppose that:

(Smoothness.) $α > 0$ , $β > 0$ , and $s > 0$ .
(Regime constants.) $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , and the window $[t_{0} - ε_{0}, t_{0} + ε_{0}]$ is contained in $(0, 1)$ .
(Smooth-covariate regime.) $d \leq 4 s$ .
(Strict-slack baseline.) Assumption 12 holds for $(d, β, s, M, c_{0}, ε_{0}, t_{0})$ .

Then there exists a constant $c > 0$ such that, for all sufficiently large $n$ , $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}),$ where $R_{n}$ is the minimax risk of Definition 3 over the same-class model of Definition 2. Moreover, $n^{- 2 α / (2 α + 1)} = ρ_{n},$ with $ρ_{n}$ as in Definition 4.

⊢ Lean

Theorem 2 packages the same conclusion in the smooth-covariate regime: it establishes a lower bound for Definition 2 and an exact exponent comparison with $ρ_{n}$ .

When $4 s < d$ , the published benchmark is governed by the covariate-smoothness term. The lower bound remains on the treatment-regression scale, and the benchmark exponent is strictly smaller.

Theorem 3 [thm:frontier-bracket-deficient] (Deficient Frontier Bracket).

Fix a covariate dimension $d \in N$ and constants $α, β, s, M, c_{0}, ε_{0}, t_{0} \in R$ . Assume:

(Regime constants.) $α > 0$ , $β > 0$ , $s > 0$ , $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , and the window $[t_{0} - ε_{0}, t_{0} + ε_{0}]$ lies in $(0, 1)$ .
(Deficient covariate smoothness.) $4 s < d$ .
(Strict-slack baseline.) Assumption 12 holds for $(d, β, s, M, c_{0}, ε_{0}, t_{0})$ .

Then there exists a constant $c > 0$ such that, for all sufficiently large sample sizes $n$ , $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}),$ where the minimax risk is as in Definition 3 and the class is Definition 2. Moreover, for the published HOIF benchmark of Definition 4, $ρ_{n} = n^{- 2/ (1 + d / (4 s) + 1/ α)}$ and its exponent is strictly smaller than the certified lower-bound exponent: $\frac{2}{1 + d / ( 4 s ) + 1/ α} < \frac{2 α}{2 α + 1} .$

⊢ Lean

Thus, in the low-covariate-smoothness regime, Theorem 3 determines the strict ordering between the same-class lower floor and the exponent in the published benchmark sequence. The same-class upper frontier remains open among the benchmark rate, an intermediate rate, and a rate depending more directly on $β$ .

For orientation, we record the external higher-order influence-function result used as a literature comparator. The cited bound from Bonvini et al. (2022) is stated over an explicitly defined, more regular subclass of the Hölder dose class in Definition 2 and supplies the corresponding upper-side benchmark.

Cited result 1 [lem:published-upper-bound-cited] (Published HOIF Comparator).

For covariate dimension $d$ and parameters $α, β, s, γ_{1}, γ_{2}$ , consider any Bonvini–Kennedy HOIF estimator specification $E$ consisting of an observed-data law $P$ , evaluation dose $t_{0}$ , nuisance estimators $\overset{μ}{^}_{n}, \overset{π}{^}_{n}, \overset{p}{^}_{n}$ , kernel $K$ , bandwidths $h_{n}$ , projection dimensions $k_{n}$ , and true and estimated projection kernels $Π_{n}$ and $Π_{n}$ . Suppose the following conditions of Bonvini et al. (2022) hold for these same inputs: $P$ is a probability law; $μ_{P}$ , $p_{X, P}$ , and $π_{P}$ are respectively the outcome regression, covariate density, and conditional treatment density of $P$ ; there exist $0 < ℓ \leq u$ such that, for every $n, a, x$ , $ℓ \leq π_{P} (a ∣ x) \leq u$ and $ℓ \leq \overset{π}{^}_{n} (a ∣ x) \leq u$ ; there exists $B > 0$ such that $∣ Y ∣ \leq B$ and $∣ A ∣ \leq B$ $P$ -almost surely and $∣ \overset{μ}{^}_{n} (a, x) ∣ \leq B$ for every $n, a, x$ ; and, for every $n$ and $x$ , the maps $a \mapsto μ_{P} (a, x)$ and $a \mapsto \overset{μ}{^}_{n} (a, x)$ lie in the one-dimensional Hölder ball of order $α$ and radius $1$ on $R$ , while $a \mapsto π_{P} (a ∣ x)$ and $a \mapsto \overset{π}{^}_{n} (a ∣ x)$ lie in the corresponding ball of order $β$ . Writing $K_{h_{n}, t_{0}} (a) = h_{n}^{- 1} K ((a - t_{0}) / h_{n})$ , assume also that there exist $K_{m a x} > 0$ and $0 < g_{ℓ} \leq g_{u}$ such that $∣ K (u) ∣ \leq K_{m a x}$ , $supp (K) \subseteq [- 1, 1]$ , $\int K (u) d u = 1$ , $\int u^{j} K (u) d u = 0$ for $1 \leq j \leq ⌈ α ⌉ - 1$ , $h_{n} > 0$ , $\int K_{h_{n}, t_{0}} (a) d a = 1$ , and, for every $n, x$ , $g_{ℓ} \leq \int K_{h_{n}, t_{0}} (a) π_{P} (a ∣ x) p_{X, P} (x) d a \leq g_{u}$ ; that there exists $C_{Π} > 0$ with $Π_{n} (x, x) \leq C_{Π} k_{n}$ and $Π_{n} (x, x) \leq C_{Π} k_{n}$ for every $n, x$ ; and that there exist $0 < r_{ℓ} \leq r_{u}$ such that, for every $n, x$ , $r_{ℓ} \leq \frac{\int K _{h_{n}, t_{0}} ( a ) π _{P} ( a ∣ x ) p _{X, P} ( x ) d a}{\int K _{h_{n}, t_{0}} ( a ) p ^ _{n} ( a , x ) d a} \leq r_{u} .$ Assume further that $d > 0$ , $α > 0$ , $β > 0$ , $s > 0$ , $α \leq β$ , $γ_{1} = γ_{2} = s$ , and that there exists $B_{X} > 0$ such that, for every $a$ and $n$ , the maps $x \mapsto μ_{P} (a, x)$ , $x \mapsto \overset{μ}{^}_{n} (a, x)$ , $x \mapsto π_{P} (a ∣ x)$ , and $x \mapsto \overset{π}{^}_{n} (a ∣ x)$ lie in the $s$ -Hölder ball of radius $B_{X}$ on $[0, 1]^{d}$ . Let $v_{n} (x) = \overset{μ}{^}_{n} (t_{0}, x) - μ_{P} (t_{0}, x)$ and $q_{n} (x) = \overset{π}{^}_{n} (t_{0} ∣ x)^{- 1} - π_{P} (t_{0} ∣ x)^{- 1}$ . The rate specialization requires $∥ v_{n} - Π_{n} v_{n} ∥_{L^{2} ([0, 1]^{d})} ∥ q_{n} - Π_{n} q_{n} ∥_{L^{2} ([0, 1]^{d})} ≲ k_{n}^{- (γ_{1} + γ_{2}) / d}$ eventually, $∥ v_{n} ∥_{L^{2} ([0, 1]^{d})} ≍ ∥ q_{n} ∥_{L^{2} ([0, 1]^{d})}$ eventually, and $(∥ v_{n} ∥_{L^{2} ([0, 1]^{d})} ∥ q_{n} ∥_{L^{2} ([0, 1]^{d})} ∥ \overset{p}{^}_{n} (t_{0}, \cdot) - π_{P} (t_{0} ∣ \cdot) p_{X, P} (\cdot) ∥_{\infty, [0, 1]^{d}})^{2} ≲ ρ_{n}$ eventually. The tuning satisfies, when $d \leq 4 s$ , $h_{n} ≍ n^{- 1/ (2 α + 1)}$ and $k_{n} ≍ n h_{n}$ eventually, and, when $4 s < d$ , $h_{n} ≍ n^{- 4 s / (α (4 s + d) + 4 s)}$ and $k_{n} ≍ (n h_{n})^{2 d / (d + 4 s)}$ eventually. Then the specified Bonvini–Kennedy HOIF estimator $\hat{θ}_{n}$ , with its nuisance-training outputs held fixed, attains the published benchmark conditional mean-squared-error rate of Definition 4: there exists $C > 0$ such that, for all sufficiently large $n$ , $\int (\hat{θ}_{n} (O_{1}, \dots, O_{n}) - θ_{P} (t_{0}))^{2} d P^{n} \leq C ρ_{n}, ρ_{n} = max {n^{- 2 α / (2 α + 1)}, n^{- 2/ (1 + d / (4 s) + 1/ α)}} .$

⊢ Lean

The notation inside the cited display belongs to the imported comparator. In the present paper, it motivates the sequence $ρ_{n}$ and provides an upper-side benchmark from a strictly smaller, more regular class; Definition 3 reserves $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0})$ for the same-class risk.

The final two remarks summarize the established rate comparison and identify the remaining same-class upper frontier.

Remark 1 [def:beta-frontier-handle] (Beta frontier handle

B_{α, β, s}

The beta comparison handle $B_{α, β, s}$ denotes the following frontier statement. For every $β > 0$ , there is a constant $c > 0$ such that, for all sufficiently large $n$ , $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) .$ For every $n \geq 1$ , when $d \leq 4 s$ , the published benchmark satisfies $ρ_{n} = n^{- 2 α / (2 α + 1)} .$ For every $n \geq 1$ , when $0 < s$ and $4 s < d$ , the published benchmark satisfies $ρ_{n} = n^{- 2/ (1 + d / (4 s) + 1/ α)}$ and its exponent obeys $\frac{2}{1 + d / ( 4 s ) + 1/ α} < \frac{2 α}{2 α + 1} .$ The upper-frontier alternatives are recorded as the following disjunction: either there is a constant $C > 0$ such that, for all sufficiently large $n$ , $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) \leq C ρ_{n},$ or there are $κ$ and $C > 0$ with $\frac{2}{1 + d / ( 4 s ) + 1/ α} < κ < \frac{2 α}{2 α + 1}$ such that, for all sufficiently large $n$ , $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) \leq C n^{- κ},$ or there are $β^{'} > 0$ , $β^{'} \neq = β$ , exponents $κ, κ^{'}$ with $κ^{'} \neq = κ$ , and positive constants $c, C, c^{'}, C^{'}$ such that, for all sufficiently large $n$ , $c n^{- κ} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) \leq C n^{- κ}$ and $c^{'} n^{- κ^{'}} \leq R_{n} (P_{α, β^{'}, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) \leq C^{'} n^{- κ^{'}} .$

⊢ Lean

Remark 1 is an interpretive shorthand for the preceding lower-bound and comparison statements. The word “frontier” denotes this exponent-level comparison and highlights the benchmark-aligned smooth-covariate regime.

Discussion, limitations, and extensions

The lower-bound analysis isolates one obstruction that is already present in the standard interior partial mean problem. The target in Definition 1 is an observed-data partial mean at a fixed interior dose, and the risk in Definition 3 is taken over the same Hölder dose class of Definition 2. Within that class, Theorem 1 gives a treatment-regression lower floor. The comparison statements in Theorem 2 and Theorem 3 relate this floor to the published benchmark sequence $ρ_{n}$ from Definition 4, while Cited result 1 supplies the external Bonvini–Kennedy comparator on its localized-regularity class.

For each fixed $β > 0$ , under the corresponding slack-baseline condition in Assumption 12, the exponent of the same-class lower bound is invariant to the treatment-density smoothness index. The constant in the lower bound, and the feasibility of the slack baseline itself, may depend on $β$ and on the other fixed model constants. Thus the conclusion is exponent-level insensitivity to $β$ , with constants calibrated separately for each fixed smoothness value.

This distinction matters most when $4 s < d$ . In that regime, Theorem 3 shows that the published benchmark sequence has a smaller exponent than the same-class lower-bound exponent. The unresolved same-class question is whether the upper behavior in low covariate smoothness follows the published benchmark, an intermediate exponent, or a rate that depends more directly on the smoothness of the treatment density.

The scope is tied to the interior positivity regime. Assumption 4 imposes a fixed lower bound on the treatment density in a neighborhood of $t_{0}$ , and Assumption 3 selects an interior evaluation window. These restrictions match the usual interior continuous-exposure formulation. Weak-overlap, design-adaptive, and trimmed targets lead to a different statistical experiment in which the amount of information near $t_{0}$ enters the rate calculation directly. Recent work on continuous exposures and overlap-sensitive causal estimands emphasizes these complementary regimes (Hejazi et al., 2022; Doss et al., 2022; Hudson et al., 2023; Shi et al., 2024; Zhang et al., 2024; Schindl et al., 2024; Bonvini et al., 2024).

Neighboring estimands raise a similar qualification. The partial mean studied here evaluates the regression surface at a single interior dose and averages over the marginal law of $X$ . Other continuous-exposure targets average over dose neighborhoods, consider stochastic interventions, or target modified exposure policies. Their definitions can smooth the treatment coordinate or alter the role of the conditional treatment density. The present lower bound is consequently a benchmark for the fixed-dose interior partial mean, while those neighboring functionals invite estimand-specific minimax analyses.

Finally, the comparison with higher-order influence-function methods is a comparison across stated model classes. The imported result in Cited result 1 supplies an upper-side benchmark under additional localized regularity, while the lower bound here is proved over the Hölder class in Definition 2. Two routes can bridge the statements: a same-class upper bound under Definition 2, or an inclusion argument covering the present Hölder class by the localized-regularity class with the needed constants.

Appendices

auxiliary lemmas and proofs

This appendix collects the auxiliary mathematical statements used to support the lower-bound and rate-comparison results in Theorem 1–Theorem 3. The statements are organized according to their role: first the two-point testing ingredients, then the algebra for the benchmark sequence $ρ_{n}$ , and finally the combined lower-bound comparison used to package the main conclusions. Throughout this appendix, $KL (Q_{0}, Q_{1})$ denotes the Kullback–Leibler divergence from $Q_{0}$ to $Q_{1}$ .

The testing reduction begins with a bounded binary outcome channel. It is convenient because the least favorable alternatives only need to separate conditional means while keeping outcomes inside the common envelope. The following elementary inequality controls the information distance between two such channels by the squared separation of their means.

Lemma 1 [lem:bernoulli-mean-channel-kl] (Bernoulli mean-channel KL bound).

Fix $B > 0$ . Let $Q_{u}$ and $Q_{v}$ be distributions on ${- B, B}$ with means $u$ and $v$ , respectively, so that $Q_{u} (B) = \frac{1 + u / B}{2}, Q_{v} (B) = \frac{1 + v / B}{2} .$ If $∣ u ∣ \leq B /2$ and $∣ v ∣ \leq B /2$ , then $KL (Q_{u}, Q_{v}) \leq \frac{2 ( u - v ) ^{2}}{B ^{2}} .$

⊢ Lean

Proof of Lemma 1.

Throughout, $Bern (r)$ denotes the two-point law on ${0, 1}$ with $Bern (r) ({1}) = r, Bern (r) ({0}) = 1 - r .$ Set $p = \frac{1 + u / B}{2}, q = \frac{1 + v / B}{2},$ so that, by the definition of the channels in the statement, $Q_{u} ({B}) = p, Q_{u} ({- B}) = 1 - p, Q_{v} ({B}) = q, Q_{v} ({- B}) = 1 - q .$

Step 1 (the two success probabilities lie in the central band). Since $B > 0$ , $p - \frac{1}{2} = \frac{u}{2 B}, q - \frac{1}{2} = \frac{v}{2 B},$ and the hypotheses $∣ u ∣ \leq B /2$ and $∣ v ∣ \leq B /2$ give $∣ u / (2 B) ∣ \leq 1/4$ and $∣ v / (2 B) ∣ \leq 1/4$ . Hence $\frac{1}{4} \leq p \leq \frac{3}{4}, \frac{1}{4} \leq q \leq \frac{3}{4} .$

Step 2 (transport to the ${0, 1}$ parametrization). Let $T (x) = 2 B x - B .$ Because $B > 0$ , the map $T$ is an affine bijection of the real line whose inverse is again affine, hence measurable, and $T (0) = - B$ , $T (1) = B$ . Consequently the image of $Bern (p)$ under $T$ puts mass $p$ at $B$ and mass $1 - p$ at $- B$ , which is exactly $Q_{u}$ ; the same computation with $q$ in place of $p$ gives $Q_{v}$ as the image of $Bern (q)$ : $Q_{u} = Bern (p) \circ T^{- 1}, Q_{v} = Bern (q) \circ T^{- 1} .$

Applying one and the same measurable bijection to both arguments leaves the Kullback–Leibler divergence unchanged, since the Radon–Nikodym derivative of the two image laws is the transport of the original one along $T$ . Therefore $KL (Q_{u}, Q_{v}) = KL (Bern (p), Bern (q)) .$

Step 3 (quadratic band bound for the Bernoulli divergence). For $p, q \in (0, 1)$ the divergence of two-point laws is the explicit sum $KL (Bern (p), Bern (q)) = p lo g \frac{p}{q} + (1 - p) lo g \frac{1 - p}{1 - q},$ and on the central band this is dominated by a quadratic: $p lo g \frac{p}{q} + (1 - p) lo g \frac{1 - p}{1 - q} \leq 4 (p - q)^{2} whenever p, q \in [\frac{1}{4}, \frac{3}{4}] .$ Indeed, writing $D (t) = t lo g t + (1 - t) lo g (1 - t)$ for the negative Bernoulli entropy, the left-hand side is the Bregman remainder $D (p) - D (q) - D^{'} (q) (p - q)$ , so the assertion is that the function $H_{q} (t) = 4 (t - q)^{2} + D^{'} (q) (t - q) + D (q) - D (t)$ is nonnegative at $t = p$ ; and it is, because $H_{q} (q) = 0$ , $H_{q}^{'} (q) = 0$ , and $H_{q}$ is convex on $[1/4, 3/4]$ , its second derivative there being $H_{q}^{''} (t) = 8 - \frac{1}{t} - \frac{1}{1 - t} \geq 8 - 4 - 4 = 0,$ so that $q$ minimizes $H_{q}$ over that interval.

Step 4 (back to the mean parametrization). Step 1 supplies the band hypothesis needed in Step 3, and $p - q = \frac{u - v}{2 B},$ so $4 (p - q)^{2} = 4 (\frac{u - v}{2 B})^{2} = \frac{( u - v ) ^{2}}{B ^{2}} \leq \frac{2 ( u - v ) ^{2}}{B ^{2}},$ the last inequality because $(u - v)^{2} / B^{2} \geq 0$ . Chaining Steps 2–4, $KL (Q_{u}, Q_{v}) \leq \frac{2 ( u - v ) ^{2}}{B ^{2}},$ which is the claim.

∎

Lemma 1 supplies the local information calculation for the outcome part of the two-point experiment. The boundedness condition $∣ u ∣, ∣ v ∣ \leq B /2$ keeps both Bernoulli probabilities uniformly away from zero and one, so the quadratic control of the divergence is uniform over the perturbations used in the construction.

The next ingredient is the standard two-point minimax testing bound. It translates a separation in the parameter into a lower bound on mean-squared error whenever the two induced sample laws remain close in information distance; this is the usual Le Cam device for nonparametric lower bounds (Tsybakov, 2009).

Lemma 2 [lem:le-cam-two-point-mse-source] (Le Cam Two-Point Bound).

Let $Q_{0}$ and $Q_{1}$ be two laws for the $n$ -sample experiment, and let $θ_{0}$ and $θ_{1}$ be the corresponding values of a real parameter. Set $Δ = ∣ θ_{1} - θ_{0} ∣.$ If $KL (Q_{0}, Q_{1}) \leq K < \infty,$ then there exists a constant $c_{K} > 0$ , depending only on $K$ , such that every estimator $T$ satisfies $i = 0, 1 max E_{Q_{i}} [(T - θ_{i})^{2}] \geq c_{K} Δ^{2} .$

⊢ Lean

Proof of Lemma 2.

Throughout the proof, $KL (Q_{0}, Q_{1})$ denotes the extended-real Kullback–Leibler divergence from $Q_{0}$ to $Q_{1}$ ; when this divergence is finite, its real value is $\int lo g (\frac{d Q _{0}}{d Q _{1}}) d Q_{0} .$ Write $TV (Q_{0}, Q_{1}) = S measurable sup Q_{0} (S) - Q_{1} (S)$ for total variation. Let $K_{+} = max {K, 0}$ and define, before any laws or estimator are chosen, $c_{K} = \frac{exp ( - K _{+} )}{32} > 0.$ Fix probability laws $Q_{0}, Q_{1}$ on a common measurable sample space satisfying the finite-budget hypothesis, and fix a measurable estimator $T$ such that the two squared losses $(T - θ_{0})^{2}$ under $Q_{0}$ and $(T - θ_{1})^{2}$ under $Q_{1}$ are integrable. Put $r = \frac{∣ θ _{1} - θ _{0} ∣}{2} = \frac{Δ}{2} \geq 0.$

Step 1 (estimation to testing). Since $2 r = ∣ θ_{0} - θ_{1} ∣$ , the two measurable error sets $S_{0} = {∣ T - θ_{0} ∣ \geq r}, S_{1} = {∣ T - θ_{1} ∣ \geq r}$ cover the sample space in the following sense: if $S_{0}$ fails, then $∣ T - θ_{0} ∣ < r$ , and the triangle inequality gives $∣ T - θ_{1} ∣ \geq ∣ θ_{0} - θ_{1} ∣ - ∣ T - θ_{0} ∣ > 2 r - r = r .$ Thus $S_{0}^{c} \subseteq S_{1}$ . The testing characterization of total variation therefore gives $Q_{0} (S_{0}) + Q_{1} (S_{1}) \geq Q_{0} (S_{0}) + Q_{1} (S_{0}^{c}) \geq 1 - TV (Q_{0}, Q_{1}) .$ Since a sum of two real numbers is at most twice their maximum, $\frac{1 - TV ( Q _{0} , Q _{1} )}{2} \leq max {Q_{0} (∣ T - θ_{0} ∣ \geq r), Q_{1} (∣ T - θ_{1} ∣ \geq r)} .$

Step 2 (a testing floor for a finite budget). The budget hypothesis is an inequality in the extended nonnegative reals against the finite value associated with the real number $K$ . Hence the divergence is finite, so $Q_{0} ≪ Q_{1}$ , and its real value satisfies $KL (Q_{0}, Q_{1}) \leq K_{+} .$ The Bretagnolle–Huber affinity bound applies under these two side conditions and yields $\frac{1}{2} exp {- KL (Q_{0}, Q_{1})} \leq 1 - TV (Q_{0}, Q_{1}) .$ Because $KL (Q_{0}, Q_{1}) \leq K_{+}$ , monotonicity of the exponential gives $exp (- K_{+}) \leq exp {- KL (Q_{0}, Q_{1})}$ . Combining this inequality with the preceding display and Step 1 gives $\frac{exp ( - K _{+} )}{4} \leq \frac{1 - TV ( Q _{0} , Q _{1} )}{2} \leq max {Q_{0} (∣ T - θ_{0} ∣ \geq r), Q_{1} (∣ T - θ_{1} ∣ \geq r)} .$

Step 3 (testing back to squared error). For $i = 0, 1$ , the event $∣ T - θ_{i} ∣ \geq r$ is the event $(T - θ_{i})^{2} \geq r^{2}$ . The squared loss is nonnegative and integrable under the corresponding law, so the integral form of Markov’s inequality gives $r^{2} Q_{i} (∣ T - θ_{i} ∣ \geq r) \leq E_{Q_{i}} {(T - θ_{i})^{2}}, i = 0, 1.$ Writing $p_{i} = Q_{i} (∣ T - θ_{i} ∣ \geq r), m_{i} = E_{Q_{i}} {(T - θ_{i})^{2}},$ and distinguishing whether $p_{0} \leq p_{1}$ or $p_{1} \leq p_{0}$ , these two inequalities imply $r^{2} max {Q_{0} (∣ T - θ_{0} ∣ \geq r), Q_{1} (∣ T - θ_{1} ∣ \geq r)} \leq i = 0, 1 max E_{Q_{i}} {(T - θ_{i})^{2}} .$

Step 4 (conclusion). Multiplying the lower bound from Step 2 by $r^{2} \geq 0$ and using Step 3, $r^{2} \frac{exp ( - K _{+} )}{4} \leq i = 0, 1 max E_{Q_{i}} {(T - θ_{i})^{2}} .$ Since $r^{2} = (θ_{1} - θ_{0})^{2} /4 = Δ^{2} /4$ , $c_{K} (θ_{1} - θ_{0})^{2} = \frac{exp ( - K _{+} )}{32} Δ^{2} \leq r^{2} \frac{exp ( - K _{+} )}{4} .$ Therefore $i = 0, 1 max E_{Q_{i}} {(T - θ_{i})^{2}} \geq \frac{exp ( - K _{+} )}{32} Δ^{2} = c_{K} Δ^{2} .$ The constant $c_{K}$ was chosen from $K$ alone, before the laws and the estimator, and the argument applies to every measurable estimator with the two stated integrability properties. This is the claimed two-point bound.

∎

Lemma 2 is used only at the level stated here. In the lower-bound construction, the alternatives are chosen so that their target values differ by $Δ$ of the desired order, while the product-law divergence is bounded by a fixed constant $K$ . The resulting mean-squared error lower bound is therefore proportional to $Δ^{2}$ .

We next record the algebraic reductions for the comparison sequence $ρ_{n}$ from Definition 4. These statements identify exactly which exponent appears in the published benchmark under the two covariate-smoothness regimes.

Lemma 3 [lem:rho-oracle-regime-algebra] (Smooth-Covariate Rate Collapse).

Let $ρ_{n}$ be the published HOIF benchmark of Definition 4, evaluated at $(n, α, s, d)$ . Suppose that

(Positive treatment smoothness.) $α > 0$ .
(Positive covariate smoothness.) $s > 0$ .
(Smooth-covariate regime.) $d \leq 4 s$ .
(Sample size.) $n \geq 1$ .

Then $ρ_{n} = n^{- 2 α / (2 α + 1)} .$

⊢ Lean

Proof of Lemma 3.

Throughout, $ρ_{n}$ is the maximum of the two power terms in Definition 4, so the claim is that the first term dominates.

Case $n = 1$ . Every real power of $1$ equals $1$ , so both terms coincide, $1^{- 2 α / (2 α + 1)} = 1 = 1^{- 2/ (1 + d / (4 s) + 1/ α)},$ and hence $ρ_{1} = max {1, 1} = 1 = 1^{- 2 α / (2 α + 1)}$ , which is the asserted identity.

Case $n > 1$ . Since $s > 0$ we have $4 s > 0$ , so the smooth-covariate hypothesis $d \leq 4 s$ can be divided through: $\frac{d}{4 s} \leq 1.$ Adding $1 + 1/ α$ to both sides, $1 + \frac{d}{4 s} + \frac{1}{α} \leq 2 + \frac{1}{α} .$ Both sides are strictly positive, since $d / (4 s) \geq 0$ and $1/ α > 0$ by $α > 0$ . Dividing the constant $2 > 0$ by the two sides therefore reverses the inequality: $\frac{2}{2 + 1/ α} \leq \frac{2}{1 + d / ( 4 s ) + 1/ α} .$

Multiplying numerator and denominator of the left-hand side by $α > 0$ gives the elementary identity $\frac{2}{2 + 1/ α} = \frac{2 α}{2 α + 1},$ so that $\frac{2 α}{2 α + 1} \leq \frac{2}{1 + d / ( 4 s ) + 1/ α} .$ Since $n > 1$ , the map $x \mapsto n^{x}$ is strictly increasing; negating the two exponents reverses the last inequality and then applying $x \mapsto n^{x}$ preserves it, so $n^{- 2/ (1 + d / (4 s) + 1/ α)} \leq n^{- 2 α / (2 α + 1)} .$ Thus the maximum in Definition 4 is attained by the treatment-regression term, that is, $ρ_{n} = n^{- 2 α / (2 α + 1)} .$

∎

Lemma 3 states that, when $d \leq 4 s$ , the maximum defining $ρ_{n}$ is attained by the treatment-regression term. This is the algebra behind the smooth-covariate comparison in Proposition 1 and Theorem 2.

Lemma 4 [lem:rho-deficient-regime-algebra] (Deficient Regime Algebra).

For any sample size $n$ , treatment smoothness $α$ , covariate smoothness $s$ , and covariate dimension $d$ , suppose that

(Positive treatment smoothness.) $α > 0$ ;
(Positive covariate smoothness.) $s > 0$ ;
(Deficient-covariate regime.) $4 s < d$ ;
(Nonzero sample size.) $n \geq 1$ .

Then the published HOIF benchmark $ρ_{n}$ from Definition 4 satisfies $ρ_{n} = n^{- 2/ (1 + d / (4 s) + 1/ α)},$ and the deficient-regime exponent is strictly smaller than the oracle exponent: $\frac{2}{1 + d / ( 4 s ) + 1/ α} < \frac{2 α}{2 α + 1} .$

⊢ Lean

Proof of Lemma 4.

Step 1 (strict comparison of the two exponents). Since $s > 0$ we have $4 s > 0$ , so the deficient-regime hypothesis $4 s < d$ can be divided through: $1 < \frac{d}{4 s} .$ Adding $1 + 1/ α$ to both sides, $2 + \frac{1}{α} < 1 + \frac{d}{4 s} + \frac{1}{α} .$ Both sides are strictly positive, because $1/ α > 0$ by $α > 0$ . Dividing the constant $2 > 0$ by the two sides therefore reverses the strict inequality: $\frac{2}{1 + d / ( 4 s ) + 1/ α} < \frac{2}{2 + 1/ α} .$ Multiplying numerator and denominator by $α > 0$ gives the elementary identity $\frac{2}{2 + 1/ α} = \frac{2 α}{2 α + 1},$ and hence the strict exponent comparison asserted in the statement, $\frac{2}{1 + d / ( 4 s ) + 1/ α} < \frac{2 α}{2 α + 1} .$

Step 2 (identification of the maximum in $ρ_{n}$ ). If $n = 1$ , every real power of $1$ equals $1$ , so both terms of Definition 4 coincide, $1^{- 2 α / (2 α + 1)} = 1 = 1^{- 2/ (1 + d / (4 s) + 1/ α)},$ and $ρ_{1} = max {1, 1} = 1 = 1^{- 2/ (1 + d / (4 s) + 1/ α)}$ , as claimed. If instead $n > 1$ , the map $x \mapsto n^{x}$ is strictly increasing; negating the two exponents reverses the strict inequality of Step 1, and applying $x \mapsto n^{x}$ then preserves the reversed inequality, so $n^{- 2 α / (2 α + 1)} \leq n^{- 2/ (1 + d / (4 s) + 1/ α)} .$ In either case the maximum in Definition 4 is attained by the covariate-smoothness term, that is, $ρ_{n} = n^{- 2/ (1 + d / (4 s) + 1/ α)} .$ Together with the strict comparison of Step 1, this is the desired conclusion.

∎

Lemma 4 gives the complementary algebra for $4 s < d$ . In that regime the published benchmark is governed by the covariate-smoothness expression, and its exponent is smaller than the exponent appearing in the same-class lower floor. The lemma thereby supplies the deficient-regime exponent comparison.

The lower-bound input for the main results is the oracle floor obtained by perturbing the treatment regression while keeping the density components within the slack baseline introduced in Assumption 12. The construction is the usual local nonparametric testing argument in the treatment coordinate (Stone, 1982; Tsybakov, 2009; Goldenshluger et al., 2020), specialized to the interior partial-mean functional.

Lemma 5 [lem:oracle-dose-regression-lower-all-beta] (All-Beta Oracle Floor).

Fix a covariate dimension $d \in N$ and constants $α, β, s, M, c_{0}, ε_{0}, t_{0} \in R$ . Suppose that:

(Positive smoothness.) $α > 0$ , $β > 0$ , and $s > 0$ .
(Regime constants.) $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , and the window of radius $ε_{0}$ around $t_{0}$ lies in the interior dose region.
(Slack baseline.) The strict-slack baseline condition of Assumption 12 holds for $(d, β, s, M, c_{0}, ε_{0}, t_{0})$ .

Then there exists a constant $c_{or} > 0$ such that, for all sufficiently large $n \in N$ , $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) \geq c_{or} n^{- 2 α / (2 α + 1)} .$ Here $P_{α, β, s} (M, c_{0}, ε_{0}, t_{0})$ is the model class of Definition 2, and $R_{n}$ is the minimax risk of Definition 3.

⊢ Lean

Proof of Lemma 5.

Step 1 (baseline, outcome scale, and the reference bump). By the strict-slack baseline condition of Assumption 12, fix a covariate density $p_{0}$ on $[0, 1]^{d}$ , a conditional treatment density $q_{0}$ on $[0, 1]$ , and a slack $η_{0} > 0$ with the properties listed there. Taking the order- $0$ bound in the Hölder condition on $q_{0}$ and combining it with $q_{0} \geq 0$ gives, in particular, $0 \leq q_{0} (a) \leq M - η_{0} for every a \in [t_{0} - ε_{0}, t_{0} + ε_{0}],$ so $M - η_{0} \geq 0$ . Put $B = \frac{M}{2} > 0.$ Let $ψ : R \to R$ be a fixed smooth reference bump with $0 \leq ψ \leq 1, ψ (z) = 1 for ∣ z ∣ \leq \frac{1}{2}, ψ (z) = 0 for ∣ z ∣ \geq 1;$ in particular $ψ (0) = 1$ , $∣ ψ ∣ \leq 1$ , and $ψ$ has compact support, so all of its derivatives are bounded.

Step 2 (the amplitude gate). There is an amplitude $λ$ with $0 < λ \leq \frac{M}{4} = \frac{B}{2}$ such that, for every sign $ζ \in {- 1, 1}$ and every bandwidth $h \in (0, 1]$ , the scaled signed bump lies in the treatment-direction Hölder ball required by Definition 2: $(a \mapsto ζ λ h^{α} ψ (\frac{a - t _{0}}{h})) \in H^{α} (M; [t_{0} - ε_{0}, t_{0} + ε_{0}]) .$ Indeed, write $k = ⌈ α ⌉ - 1$ for the largest integer strictly below $α$ , let $C_{j} = sup_{z} ∣ ψ^{(j)} (z) ∣ < \infty$ for $j \leq k$ , let $C_{H}$ be an $(α - k)$ -Hölder constant of $ψ^{(k)}$ on $R$ , and set $λ = min {\frac{M}{4}, \frac{M}{1 + max { C _{H} , C _{0} + \dots + C _{k} , 1 }}} > 0.$ For $j \leq k$ the $j$ -th derivative of the scaled bump at $a$ is $ζ λ h^{α - j} ψ^{(j)} ((a - t_{0}) / h)$ , and $h^{α - j} \leq 1$ because $0 < h \leq 1$ and $j \leq k \leq α$ ; hence its modulus is at most $λ C_{j} \leq M$ . For the top order, writing $u_{x} = (x - t_{0}) / h$ and using $∣ u_{x} - u_{y} ∣ = ∣ x - y ∣/ h$ , $ζ λ h^{α - k} {ψ^{(k)} (u_{x}) - ψ^{(k)} (u_{y})} \leq λ h^{α - k} C_{H} \frac{∣ x - y ∣ ^{α - k}}{h ^{α - k}} = λ C_{H} ∣ x - y ∣^{α - k} \leq M ∣ x - y ∣^{α - k},$ which is exactly the top-order Hölder requirement of radius $M$ .

Step 3 (the divergence budget and the constant). Set $K = \frac{16 λ ^{2} ( M - η _{0} )}{B ^{2}},$ let $c_{K} > 0$ be the constant supplied by Lemma 2 for this budget, and define $c_{or} = 4 c_{K} λ^{2} > 0.$

Step 4 (bandwidth). For $n \geq 1$ put $h = h_{n} = n^{- 1/ (2 α + 1)} .$ Since $α > 0$ we have $2 α + 1 > 0$ , so $h_{n} \to 0$ ; hence for all sufficiently large $n$ both $0 < h \leq 1$ and $h \leq ε_{0}$ hold. Moreover, for every $n \geq 1$ , $n h^{2 α + 1} = n \cdot n^{- (2 α + 1) / (2 α + 1)} = n \cdot n^{- 1} = 1.$ Fix such an $n$ from here on.

Step 5 (the two least favourable laws). For $ζ \in {- 1, 1}$ define the perturbed treatment regression $μ_{ζ} (a, x) = ζ λ h^{α} ψ (\frac{a - t _{0}}{h}), a \in [0, 1], x \in [0, 1]^{d},$ and let $P_{ζ}$ be the observed-data law generated as follows: draw $X$ on $[0, 1]^{d}$ with density $p_{0}$ ; independently draw $A$ on $[0, 1]$ with density $q_{0}$ ; and, given $(A, X) = (a, x)$ , draw $Y$ from the two-point channel $Q_{μ_{ζ} (a, x)}$ on ${- B, B}$ of Lemma 1. The potential-outcome overlay is $Y (a) = {Y, μ_{ζ} (a, X), A = a, A \neq = a .$ By construction the law-side nuisances of $P_{ζ}$ are $μ_{P_{ζ}} (a, x) = μ_{ζ} (a, x), π_{P_{ζ}} (a ∣ x) = q_{0} (a), p_{X, P_{ζ}} = p_{0} .$ Write $P_{0} = P_{- 1}$ and $P_{1} = P_{+ 1}$ . Because $∣ ζ ∣ = 1$ , $0 \leq ψ \leq 1$ , $0 < h \leq 1$ and $α > 0$ (so $h^{α} \leq 1$ ), $∣ μ_{ζ} (a, x) ∣ = λ h^{α} ψ (\frac{a - t _{0}}{h}) \leq λ \leq \frac{B}{2},$ so in particular $∣ μ_{ζ} ∣ \leq B$ and each channel $Q_{μ_{ζ} (a, x)}$ is a genuine probability measure on ${- B, B}$ .

Step 6 (both laws lie in the model class). We check the conditions listed in Definition 2 for $P_{ζ}$ , $ζ \in {- 1, 1}$ .

Sampling. $p_{0} \geq 0$ with $\int_{[0, 1]^{d}} p_{0} = 1$ , $q_{0} \geq 0$ with $\int_{0}^{1} q_{0} = 1$ , and each outcome channel is a probability measure by the last display of Step 5; so $P_{ζ}$ is a probability law under which $A \in [0, 1]$ and $X \in [0, 1]^{d}$ almost surely, and the sample is $n$ independent draws from it.
Bounded outcome. $Y \in {- B, B}$ , so $∣ Y ∣ = B = M /2 \leq M$ almost surely.
Interior dose. $[t_{0} - ε_{0}, t_{0} + ε_{0}] \subset (0, 1)$ is a hypothesis of the present lemma.
Local positivity. $π_{P_{ζ}} (a ∣ x) = q_{0} (a) \geq c_{0} + η_{0} \geq c_{0}$ for every $a$ in the window and every $x \in [0, 1]^{d}$ , by Assumption 12 and $η_{0} > 0$ .
Treatment-direction smoothness of the regression. This is exactly the conclusion of Step 2, since $μ_{P_{ζ}} (\cdot, x)$ does not depend on $x$ .
Treatment-direction smoothness of the density. $q_{0}$ has Hölder norm at most $M - η_{0}$ of order $β$ on the window, and $M - η_{0} \leq M$ , so $q_{0} \in H^{β} (M; [t_{0} - ε_{0}, t_{0} + ε_{0}])$ .
Covariate-direction smoothness of the regression. Since $ψ (0) = 1$ , $μ_{P_{ζ}} (t_{0}, x) = ζ λ h^{α} for every x \in [0, 1]^{d},$ a constant function of $x$ of modulus at most $B \leq M$ by Step 5; a constant of modulus at most $M$ lies in $H^{s} (M; [0, 1]^{d})$ , all its derivatives vanishing.
Covariate-direction smoothness of the density. Likewise $π_{P_{ζ}} (t_{0} ∣ x) = q_{0} (t_{0})$ is constant in $x$ , of modulus at most $M - η_{0} \leq M$ , hence in $H^{s} (M; [0, 1]^{d})$ .
Covariate density. $p_{0} \in H^{s} (M - η_{0}; [0, 1]^{d}) \subseteq H^{s} (M; [0, 1]^{d})$ , and $0 \leq p_{0} \leq M - η_{0} \leq M$ on the cube.
Semantic ties. The channel $Q_{u}$ has mean $u$ , so $E_{P_{ζ}} [Y ∣ A = a, X = x] = μ_{ζ} (a, x) = μ_{P_{ζ}} (a, x)$ ; the $X$ -marginal of $P_{ζ}$ is $p_{0}$ times Lebesgue measure on $[0, 1]^{d}$ ; and the $(A, X)$ -law has Lebesgue density $(a, x) \mapsto q_{0} (a) p_{0} (x) = π_{P_{ζ}} (a ∣ x) p_{X, P_{ζ}} (x)$ on $[0, 1] \times [0, 1]^{d}$ , which is exactly the required factorization.
Causal overlay. Consistency $Y = Y (A)$ holds by the definition of $Y (\cdot)$ in Step 5. For ignorability, fix $a$ ; because $A$ has a Lebesgue density the event ${A = a}$ is null, so $Y (a) = μ_{ζ} (a, X)$ almost surely, a deterministic function of $X$ alone; hence $Y (a)$ is conditionally independent of $A$ given $X$ .

Therefore $P_{0}, P_{1} \in P_{α, β, s} (M, c_{0}, ε_{0}, t_{0})$ .

Step 7 (one-observation divergence). The two laws share the same $(A, X)$ -law, with density $q_{0} (a) p_{0} (x)$ , and differ only in the conditional outcome channel; since the observed pair $(A, X)$ is a function of the observation and each fibre channel is dominated by its counterpart, the chain rule for divergence gives $KL (P_{0}, P_{1}) = \int_{[0, 1]^{d}} \int_{0}^{1} KL (Q_{μ_{- 1} (a, x)}, Q_{μ_{1} (a, x)}) q_{0} (a) p_{0} (x) d a d x .$ Fibrewise, $∣ μ_{- 1} (a, x) ∣ \leq B /2$ and $∣ μ_{1} (a, x) ∣ \leq B /2$ by Step 5, so Lemma 1 applies with $u = μ_{- 1} (a, x)$ and $v = μ_{1} (a, x)$ ; as $μ_{- 1} (a, x) - μ_{1} (a, x) = - 2 λ h^{α} ψ (\frac{a - t _{0}}{h}),$ it yields $KL (Q_{μ_{- 1} (a, x)}, Q_{μ_{1} (a, x)}) \leq \frac{2 ( μ _{- 1} ( a , x ) - μ _{1} ( a , x ) ) ^{2}}{B ^{2}} = \frac{8 λ ^{2} h ^{2 α}}{B ^{2}} ψ (\frac{a - t _{0}}{h})^{2} .$ The bump concentrates the remaining integral: $ψ ((a - t_{0}) / h) = 0$ whenever $∣ a - t_{0} ∣ \geq h$ , so the integrand vanishes outside $(t_{0} - h, t_{0} + h)$ ; on that interval, which is contained in the window because $h \leq ε_{0}$ , we have $ψ^{2} \leq 1$ and $q_{0} \leq M - η_{0}$ by Step 1; and the interval has length $2 h$ . Hence $\int_{0}^{1} ψ (\frac{a - t _{0}}{h})^{2} q_{0} (a) d a \leq 2 (M - η_{0}) h .$

Using $\int_{[0, 1]^{d}} p_{0} = 1$ , the two previous displays combine to the one-observation estimate $KL (P_{0}, P_{1}) \leq \frac{8 λ ^{2} h ^{2 α}}{B ^{2}} \cdot 2 (M - η_{0}) h = \frac{16 λ ^{2} ( M - η _{0} )}{B ^{2}} h^{2 α + 1} .$

Step 8 (tensorization and the budget). Each fibre channel puts mass at least $1/4$ on each of the two atoms $\pm B$ , because $∣ μ_{ζ} ∣ \leq B /2$ ; consequently $P_{0}$ and $P_{1}$ are mutually absolutely continuous with integrable log-likelihood ratio, and the divergence between the $n$ -fold product laws tensorizes: $KL (P_{0}^{\otimes n}, P_{1}^{\otimes n}) \leq n KL (P_{0}, P_{1}) \leq n \frac{16 λ ^{2} ( M - η _{0} )}{B ^{2}} h^{2 α + 1} = \frac{16 λ ^{2} ( M - η _{0} )}{B ^{2}} = K,$ the last equality by the calibration $n h^{2 α + 1} = 1$ of Step 4.

Step 9 (functional separation). Since $ψ (0) = 1$ and $\int_{[0, 1]^{d}} p_{0} = 1$ , $θ_{P_{ζ}} (t_{0}) = \int_{[0, 1]^{d}} μ_{ζ} (t_{0}, x) p_{0} (x) d x = ζ λ h^{α} ψ (0) \int_{[0, 1]^{d}} p_{0} (x) d x = ζ λ h^{α},$ so the two target values are separated by $θ_{P_{1}} (t_{0}) - θ_{P_{0}} (t_{0}) = 2 λ h^{α} .$

Step 10 (two-point reduction inside the minimax risk). Let $\hat{θ}_{n}$ be any estimator admissible in Definition 3, that is, measurable with values in $[- M, M]$ . For every $P$ in the model class, $∣ μ_{P} (t_{0}, \cdot) ∣ \leq M$ and $0 \leq p_{X, P} \leq M$ , and $[0, 1]^{d}$ has unit Lebesgue measure, so $∣ θ_{P} (t_{0}) ∣ = \int_{[0, 1]^{d}} μ_{P} (t_{0}, x) p_{X, P} (x) d x \leq M^{2}, hence \hat{θ}_{n} - θ_{P} (t_{0}) \leq M + M^{2} .$

Thus under every $n$ -fold product law the squared loss is bounded by $(M + M^{2})^{2}$ , so it is integrable and the worst-case risk over the class is a finite supremum. Apply Lemma 2 with the $n$ -sample laws $Q_{0} = P_{0}^{\otimes n}$ , $Q_{1} = P_{1}^{\otimes n}$ , parameter values $θ_{i} = θ_{P_{i}} (t_{0})$ , budget $K$ (available by Step 8) and estimator $T = \hat{θ}_{n}$ : $c_{K} (θ_{P_{1}} (t_{0}) - θ_{P_{0}} (t_{0}))^{2} \leq i = 0, 1 max \int (\hat{θ}_{n} - θ_{P_{i}} (t_{0}))^{2} d P_{i}^{\otimes n} .$ Both $P_{0}$ and $P_{1}$ belong to the class by Step 6, so the right-hand side is at most the supremum over $P \in P_{α, β, s} (M, c_{0}, ε_{0}, t_{0})$ of the risk of $\hat{θ}_{n}$ . As $\hat{θ}_{n}$ was arbitrary, taking the infimum over admissible estimators preserves the bound, and with the separation of Step 9, $c_{K} (2 λ h^{α})^{2} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) .$

Step 11 (conclusion). Finally, by the choice of $h$ in Step 4, $c_{K} (2 λ h^{α})^{2} = 4 c_{K} λ^{2} h^{2 α} = c_{or} (n^{- 1/ (2 α + 1)})^{2 α} = c_{or} n^{- 2 α / (2 α + 1)} .$ Combining the last two displays, $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) \geq c_{or} n^{- 2 α / (2 α + 1)}$ for all sufficiently large $n$ , with $c_{or} = 4 c_{K} λ^{2} > 0$ depending only on the fixed model constants and the slack baseline, not on $n$ .

∎

Lemma 5 is the same-class lower-bound engine. Its conclusion is uniform in the sense relevant for the paper’s exponent comparison: for each fixed admissible $β > 0$ , the lower exponent is $2 α / (2 α + 1)$ . The constant $c_{or}$ may depend on the fixed model parameters and on the slack available in Assumption 12; the stated uniformity concerns each fixed $β$ .

verification note

This appendix records the verification scope for the formal statements used in the paper. The observed-data experiment, causal overlay, Hölder dose class, target functional, minimax risk, published benchmark sequence, auxiliary testing statements, algebraic rate comparisons, and lower-bound consequences are represented by the numbered assumptions, definitions, lemmas, propositions, theorems, and remarks displayed in the paper. The corresponding theorem statements and their derivations are machine-checked in Lean 4.

The machine-checked part covers the consequences of the stated assumptions and auxiliary lemmas as they are formulated here. In particular, Assumption 1–Assumption 12 define the statistical and causal restrictions used by the displayed results; Definition 1, Definition 2, Definition 3, and Definition 4 fix the target, model class, risk, and comparison sequence; and the auxiliary appendix lemmas supply the testing and algebraic inputs used to derive Theorem 1, Theorem 2, and Theorem 3 together with Proposition 1. The verification establishes the same-class lower-bound and rate-comparison statements, with the upper frontier identified as a separate research question.

Several ingredients enter as mathematical assumptions or cited comparators. The causal identification primitives in Assumption 10 and Assumption 11 are imposed conditions on the observed-data law with its potential-outcome interpretation. The formal argument uses the classical Le Cam minimax reduction at the level of the auxiliary testing lemma stated in the appendix and verifies its consequences for the present model and rates. The Bonvini–Kennedy result in Cited result 1, imported from Bonvini et al. (2022), motivates the benchmark sequence $ρ_{n}$ on its localized-regularity class; Definition 2 specifies the distinct same-class lower-bound model studied here.

The Lean-oriented labels are therefore confined to cross-references and this reproducibility note. In the main text they serve only to identify the precise assumptions and statements being invoked. The econometric content of the paper is the lower bound over Definition 2, the algebraic comparison with Definition 4, and the explicit separation between those same-class lower results and the external higher-order influence-function comparator.

Proofs of the main results

Proof of Theorem 1.

Instantiate the oracle floor. The theorem hypotheses provide the inputs required by Lemma 5 for the same covariate dimension $d$ and the same constants $α, β, s, M, c_{0}, ε_{0}, t_{0}$ . The positive-smoothness assumptions give $α > 0$ , $β > 0$ , and $s > 0$ . The regime-constants assumptions give $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , and the interior-window condition recorded in Assumption 3. The strict-slack baseline assumption is Assumption 12 for the tuple $(d, β, s, M, c_{0}, ε_{0}, t_{0})$ , matching the slack-baseline input of the lemma.
Apply the oracle floor. Applying Lemma 5 with these inputs yields a constant $c_{or} > 0$ such that, for all sufficiently large $n \in N$ , $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) \geq c_{or} n^{- 2 α / (2 α + 1)} .$ Here the class is the model class of Definition 2, and $R_{n}$ is the minimax risk of Definition 3. The displayed tail bound is the conclusion of the cited oracle floor for the fixed treatment-density smoothness parameter $β > 0$ .
Conclusion. Set $c = c_{or}$ . Then $c > 0$ , and the inequality from Step 2 is equivalently $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0})$ for all sufficiently large sample sizes $n$ . This is the asserted positive constant and eventual lower bound over the model class of Definition 2 for the minimax risk of Definition 3.

∎

Proof of Proposition 1.

The positive-smoothness, regime-constant, and baseline-slack hypotheses assumed here – $α > 0$ , $β > 0$ , $s > 0$ , $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , $[t_{0} - ε_{0}, t_{0} + ε_{0}] \subset (0, 1)$ , and Assumption 12 for $(d, β, s, M, c_{0}, ε_{0}, t_{0})$ – are the hypotheses required by Theorem 1. Applying that result gives a constant $c > 0$ such that, for all sufficiently large sample sizes $n$ , $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) .$ Fix this $c$ for the rest of the proof.
Intersect the eventual lower-bound event supplied by Theorem 1 with the eventual sample-size event $n \geq 1$ . Equivalently, after increasing the sufficiently-large threshold if necessary, every sample size under consideration satisfies both $n \geq 1$ and the lower-bound inequality supplied by Theorem 1.
For any such $n$ , the hypotheses $α > 0$ , $s > 0$ , $d \leq 4 s$ , and $n \geq 1$ are the hypotheses of Lemma 3. Hence the published benchmark rate satisfies $ρ_{n} = n^{- 2 α / (2 α + 1)} .$
For the same $n$ , the lower-bound inequality supplied by Theorem 1 gives $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) .$ The identity supplied by Lemma 3 rewrites $c ρ_{n}$ as $c n^{- 2 α / (2 α + 1)}$ , and therefore $c ρ_{n} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) .$ This holds on the same eventual set of sample sizes. Together with $c > 0$ and the displayed identity for $ρ_{n}$ , it is the assertion of Proposition 1.

∎

Proof of Theorem 2.

The smoothness, regime-constant and strict-slack hypotheses assumed here — $α > 0$ , $β > 0$ , $s > 0$ , $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , $[t_{0} - ε_{0}, t_{0} + ε_{0}] \subset (0, 1)$ , and Assumption 12 for $(d, β, s, M, c_{0}, ε_{0}, t_{0})$ — are exactly the hypotheses of Theorem 1; the smooth-covariate condition $d \leq 4 s$ is not needed for this step and is used only in Step 3. That theorem therefore supplies a constant $c > 0$ and a threshold $n_{1}$ with $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) for every n \geq n_{1} .$ Fix this $c$ ; it is the constant asserted in the statement.
Set $n_{0} = max {n_{1}, 1}$ . For every $n \geq n_{0}$ both the displayed lower bound of Step 1 and the elementary condition $n \geq 1$ hold; this is the intersection of two eventual conditions, and the conjunction asserted by the theorem is claimed only for such $n$ .
Fix $n \geq n_{0}$ . The hypotheses $α > 0$ , $s > 0$ , $d \leq 4 s$ and $n \geq 1$ are exactly those of Lemma 3, so $ρ_{n} = n^{- 2 α / (2 α + 1)}, equivalently n^{- 2 α / (2 α + 1)} = ρ_{n},$ which is the second assertion of the theorem, in the orientation in which it is stated. Holding this identity together with the lower bound of Step 1 on the same range $n \geq n_{0}$ gives, for the same constant $c > 0$ , the asserted eventual conjunction of the lower bound and the exponent identity.

∎

Proof of Theorem 3.

The regime-constant and strict-slack hypotheses assumed here — $α > 0$ , $β > 0$ , $s > 0$ , $M > 0$ , $c_{0} > 0$ , $t_{0} \in (0, 1)$ , $ε_{0} \in (0, 1/2)$ , $[t_{0} - ε_{0}, t_{0} + ε_{0}] \subset (0, 1)$ , and Assumption 12 for $(d, β, s, M, c_{0}, ε_{0}, t_{0})$ — are exactly the hypotheses of Lemma 5; the deficient-covariate condition $4 s < d$ is not used for this step and enters only in Step 3. That lemma therefore supplies a constant $c > 0$ and a threshold $n_{1}$ with $R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}) \geq c n^{- 2 α / (2 α + 1)} for every n \geq n_{1} .$ Fix this $c$ ; it is the constant asserted in the statement. Note that the lower floor is established for the given fixed $β > 0$ and its exponent does not involve $β$ .
Set $n_{0} = max {n_{1}, 1}$ . For every $n \geq n_{0}$ both the display of Step 1 and the elementary condition $n \geq 1$ hold; this is the intersection of two eventual conditions, and the conjunction asserted by the theorem is claimed only for such $n$ .
Fix $n \geq n_{0}$ . The hypotheses $α > 0$ , $s > 0$ , $4 s < d$ and $n \geq 1$ are exactly those of Lemma 4, which gives both the deficient-regime formula $ρ_{n} = n^{- 2/ (1 + d / (4 s) + 1/ α)}$ and the strict comparison of exponents $\frac{2}{1 + d / ( 4 s ) + 1/ α} < \frac{2 α}{2 α + 1} .$
On the common range $n \geq n_{0}$ , the lower bound of Step 1, rewritten in the orientation of the statement as $c n^{- 2 α / (2 α + 1)} \leq R_{n} (P_{α, β, s} (M, c_{0}, ε_{0}, t_{0}), t_{0}),$ holds together with the two conclusions of Step 3. That conjunction, with the constant $c > 0$ fixed in Step 1, is exactly the assertion of the theorem: the certified lower floor sits at the treatment-regression exponent $2 α / (2 α + 1)$ , while the published benchmark sequence is the covariate-smoothness power, whose exponent is strictly smaller.

∎

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Hudson, Aaron and Geng, Elvin H. and Odeny, Thomas A. and Bukusi, Elizabeth A. and Petersen, Maya L. and van der Laan, Mark J. (2023). An Approach to Nonparametric Inference on the Causal Dose Response Function. . doi
Bonvini, Matteo and Kennedy, Edward H. and Dukes, Oliver and Balakrishnan, Sivaraman (2024). Doubly-Robust Inference and Optimality in Structure-Agnostic Models with Smoothness. . doi
Zhang, Yikun and Chen, Yen-Chi and Giessing, Alexander (2024). Nonparametric Inference on Dose-Response Curves Without the Positivity Condition. . doi
Shi, Junming and Zhang, Wenxin and Hubbard, Alan E. and van der Laan, Mark J. (2024). HAL-Based Plug-in Estimation with Pointwise Asymptotic Normality of the Causal Dose-Response Curve. . doi
Schindl, Kyle and Shen, Shuying and Kennedy, Edward H. (2024). Incremental Effects for Continuous Exposures. . doi

Formal layer — Lean correspondence

Every object below is formalized in Lean 4. Click a verified row to view its statement; the rest link to the Lean source.

Setups (3)

Setup S-1 — iid observational sampling: observed unit O=(Y,A,X) in R x [0,1] x [0,1]^d, single-observation data law P (a probability Measure on the unit space) with induced marginal P_X, nuisances mu_P, pi_P, p_{X,P}; n-sample law is the i.i.d. product Measure.pi (fun _ : Fin n => P). — Direct architectural analogue: Causalean.Estimation.MinimaxATE.Model (Obs/obsLaw/productLaw/minimaxMiss/nMSE/nMiss_sq_le_nMSE) realizes exactly this i.i.d. two-point world but over a FINITE covariate (X : Fin M, Y : Bool) so all laws are finite PMFs. Here X ranges over [0,1]^d and A over [0,1] with a continuous-in-treatment bump, so the single-observation law is a genuine Measure (Mathlib MeasureTheory + Measure.pi tensorization); cannot reuse the finite-PMF Model verbatim -> define-local over Mathlib, importing the Stat.Minimax Le Cam layer. Search trace: surveyed Causalean/Stat/Minimax/{LeCam,MinimaxRisk,Pinsker,ChiSquared,TotalVariation}, Estimation/MinimaxATE; no continuous-treatment dose-response sampling world present.
Setup S-2 — potential-outcome causal overlay on S1: the PO process a -> Y(a) on the same probability space, with consistency (Y = Y(A) a.s.) and conditional ignorability (Y(a) indep A | X); the causal target value E_P[Y(t_0)] is identified by the observed-data partial mean theta_P(t_0). — Causalean.PO (POSystem, POSystem.Consistency, counterfactual-independence) is the framework analogue but is a skeleton (sorry-bearing) and graph/regime-indexed, heavier than the scalar continuous-treatment overlay needed here. The construction supplies its own explicit PO realization (draw U ~ Unif[0,1], set Y_zeta(a) via the two-point mean channel), so consistency + ignorability are verified directly on the constructed laws rather than imported. Define-local over Mathlib with PO naming aligned to Causalean.PO. Search trace: surveyed Causalean/PO/{Core,Assumptions}; no continuous-dose PO partial-mean world to reuse verbatim.
Setup S-3 — regime constants and frontier objects: smoothness/positivity/bound scalars and the evaluation geometry, plus the minimax-risk and published-rate frontier sequences. — Plain real/nat regime parameters (alpha,beta,s,M,c_0,t_0,epsilon_0 : R>0, d,n : N) and two real-valued frontier sequences R_n (minimax MSE, def:minimax-risk) and rho_n (def:published-hoif-rate). No measure-theoretic world; bound directly in Basic.lean.

Definitions (5)

Definition P-1 verified CausalSmith.Stat.DoseResponseMinimax.HolderDoseClass {P : P satisfies ass:iid-sampling, ass:consistency, ass:no-unmeasured-confounding, ass:bounded-outcome, ass:interior-dose, ass:local-positivity, ass:mu-treatment-holder, ass:pi-treatment-holder, ass:mu-covariate-holder, ass:pi-covariate-holder, ass:px-holder}
Definition P-2 verified CausalSmith.Stat.DoseResponseMinimax.thetaFunctional theta_P(t_0) = int_{[0,1]^d} mu_P(t_0,x) p_{X,P}(x) dx
Definition P-3 verified CausalSmith.Stat.DoseResponseMinimax.minimaxRisk R_n(P_{alpha,beta,s}(M,c_0,epsilon_0,t_0), t_0) = inf_{hat_theta_n} sup_{P in P_{alpha,beta,s}(M,c_0,epsilon_0,t_0)} E_P[(hat_theta_n - theta_P(t_0))^2]
Definition P-4 verified CausalSmith.Stat.DoseResponseMinimax.publishedHoifRate rho_n = n^(-2*alpha/(2*alpha+1)) vee n^(-2/(1 + d/(4*s) + 1/alpha))
Definition P-5 verified CausalSmith.Stat.DoseResponseMinimax.betaFrontierHandle Headline and frontier handle. The delivered content is the all-beta lower floor: by a de-laundered two-point Le Cam construction perturbing only mu_P with the covariate density and treatment-density baselines fixed, the original class satisfies R_n(P_{alpha,beta,s}(M,c_0,epsilon_0,t_0),t_0) >= c n^{-2*alpha/(2*alpha+1)} for every beta>0. When s>=d/4, lem:rho-oracle-regime-algebra identifies the published benchmark rho_n with this same exponent, so the converse lands on the classical interior treatment-regression scale. When 0<s<d/4, rho_n = n^{-2/(1+d/(4*s)+1/alpha)} is recorded only as the published HOIF comparator, not as a same-class certified upper endpoint. The residual carried by oeq:full-beta-frontier is therefore the unrestricted upper frontier itself: can one attain rho_n on the original class, or does a genuinely beta-sensitive phase intervene?

Assumptions (12)

Assumption A-1 verified CausalSmith.Stat.DoseResponseMinimax.IidSampling O_1, ..., O_n are iid draws from P.
Assumption A-2 verified CausalSmith.Stat.DoseResponseMinimax.Consistency Y = Y(A) almost surely under P.
Assumption A-3 verified CausalSmith.Stat.DoseResponseMinimax.NoUnmeasuredConfounding For every a in [0,1], Y(a) is independent of A conditional on X under P.
Assumption A-4 verified CausalSmith.Stat.DoseResponseMinimax.BoundedOutcome |Y| <= M almost surely under P.
Assumption A-5 verified CausalSmith.Stat.DoseResponseMinimax.InteriorDose [t_0 - epsilon_0, t_0 + epsilon_0] is contained in (0,1).
Assumption A-6 verified CausalSmith.Stat.DoseResponseMinimax.LocalPositivity pi_P(a|x) >= c_0 for every a in [t_0 - epsilon_0, t_0 + epsilon_0] and every x in [0,1]^d.
Assumption A-7 verified CausalSmith.Stat.DoseResponseMinimax.MuTreatmentHolder For every x in [0,1]^d, the function a -> mu_P(a,x) belongs to a Holder ball of order alpha and radius M on [t_0 - epsilon_0, t_0 + epsilon_0].
Assumption A-8 verified CausalSmith.Stat.DoseResponseMinimax.PiTreatmentHolder For every x in [0,1]^d, the function a -> pi_P(a|x) belongs to a Holder ball of order beta and radius M on [t_0 - epsilon_0, t_0 + epsilon_0].
Assumption A-9 verified CausalSmith.Stat.DoseResponseMinimax.MuCovariateHolder The function x -> mu_P(t_0,x) belongs to a Holder ball of order s and radius M on [0,1]^d.
Assumption A-10 verified CausalSmith.Stat.DoseResponseMinimax.PiCovariateHolder The function x -> pi_P(t_0|x) belongs to a Holder ball of order s and radius M on [0,1]^d.
Assumption A-11 verified CausalSmith.Stat.DoseResponseMinimax.PxHolder The density p_{X,P} belongs to a Holder ball of order s and radius M on [0,1]^d and is bounded above by M.
Assumption A-12 verified CausalSmith.Stat.DoseResponseMinimax.BaselineSubmodelSlack There exist a density p_0 on [0,1]^d, a conditional treatment density q_0 on [0,1], and a constant eta_0>0 such that p_0 and q_0 have Holder norms at most M-eta_0 in the smoothness classes required of p_{X,P} and pi_P, q_0(a) >= c_0+eta_0 for all a in [t_0-epsilon_0,t_0+epsilon_0], and p_0 is bounded above by M-eta_0.

Lemmas (6)

Lemma L-1 verified CausalSmith.Stat.DoseResponseMinimax.bernoulli_mean_channel_kl_source Fix B>0. Let Q_u and Q_v be distributions on {-B,B} with means u and v, respectively, so Q_u(B)=(1+u/B)/2 and Q_v(B)=(1+v/B)/2. If |u|<=B/2 and |v|<=B/2, then KL(Q_u,Q_v) <= 2*(u-v)^2/B^2.
Lemma L-2 verified CausalSmith.Stat.DoseResponseMinimax.le_cam_two_point_mse_source Let Q_0 and Q_1 be two laws for the n-sample experiment and let theta_0,theta_1 be the corresponding values of a real parameter, with Delta=|theta_1-theta_0|. If KL(Q_0,Q_1)<=K<infinity, then every estimator T satisfies max_{i=0,1} E_{Q_i}[(T-theta_i)^2] >= c_K Delta^2 for a constant c_K>0 depending only on K.
Lemma L-3 verified CausalSmith.Stat.DoseResponseMinimax.rho_oracle_regime_algebra If s >= d/4, then the published beta>=alpha rate rho_n satisfies rho_n = n^{-2 alpha/(2 alpha+1)} for every n >= 1.
Lemma L-4 verified CausalSmith.Stat.DoseResponseMinimax.oracle_dose_regression_lower_all_beta Assume ass:baseline-submodel-slack. For every beta>0 there is a constant c_or>0, depending only on the fixed model radii and on the slack baseline but not on n, such that for all sufficiently large n, R_n(P_{alpha,beta,s}(M,c_0,epsilon_0,t_0),t_0) >= c_or n^{-2 alpha/(2 alpha+1)}. The construction leaves the treatment density fixed, so the bound is independent of whether beta>=alpha.
Lemma L-5 verified CausalSmith.Stat.DoseResponseMinimax.rho_deficient_regime_algebra If 0<s<d/4, then rho_n = n^{-2/(1+d/(4s)+1/alpha)} for every n>=1, and 2/(1+d/(4s)+1/alpha) < 2 alpha/(2 alpha+1).
Lemma L-6 verified CausalSmith.Stat.DoseResponseMinimax.certified_beta_frontier_bracket Assume ass:baseline-submodel-slack. For every fixed beta>0 there exists c>0 such that, for all sufficiently large n, R_n(P_{alpha,beta,s}(M,c_0,epsilon_0,t_0),t_0) >= c n^{-2 alpha/(2 alpha+1)}. If s>=d/4 then rho_n = n^{-2 alpha/(2 alpha+1)}. If 0<s<d/4 then rho_n = n^{-2/(1+d/(4s)+1/alpha)} and 2/(1+d/(4s)+1/alpha) < 2 alpha/(2 alpha+1). Hence the certified unrestricted beta-frontier is the all-beta oracle lower floor together with the algebraic comparison to the published rho_n benchmark. ↗ Lean

Theorems (5)

Theorem T-1 verified CausalSmith.Stat.DoseResponseMinimax.sharp_pointwise_lower_bound Assume ass:baseline-submodel-slack. Then for every beta>0 there exists a constant c > 0, depending only on the fixed model radii and the slack baseline, such that for all sufficiently large n, R_n(P_{alpha,beta,s}(M,c_0,epsilon_0,t_0), t_0) >= c n^(-2*alpha/(2*alpha+1)). Equivalently, uniformly over every beta>0, every estimator of theta_P(t_0) over P_{alpha,beta,s}(M,c_0,epsilon_0,t_0) has worst-case mean-squared error at least a constant multiple of the classical interior treatment-regression pointwise rate.
Theorem T-2 verified CausalSmith.Stat.DoseResponseMinimax.oracle_regime_reduction Under the assumptions of thm:sharp-pointwise-lower-bound, if s >= d/4 then rho_n = n^(-2*alpha/(2*alpha+1)) and therefore R_n(P_{alpha,beta,s}(M,c_0,epsilon_0,t_0), t_0) >= c n^(-2*alpha/(2*alpha+1)) = c rho_n for some c > 0. Thus the certified lower floor reduces to the classical interior pointwise nonparametric regression barrier.
Theorem T-3 verified CausalSmith.Stat.DoseResponseMinimax.certifiedPartialBetaFrontier Under baseline-submodel slack, for every fixed beta > 0 the original dose-response class has minimax MSE at least c n^(-2 alpha/(2 alpha+1)) for all sufficiently large n. This lower floor matches the published rho_n exponent when s >= d/4; when 0 < s < d/4, rho_n has a strictly smaller exponent and remains only an upper-side comparator, so no same-class upper endpoint is asserted. ↗ Lean
Theorem T-4 verified CausalSmith.Stat.DoseResponseMinimax.sharp_minimax_smooth_covariate Assume ass:baseline-submodel-slack. For every fixed beta>0 and every s>=d/4, there exists c>0 such that, for all sufficiently large n, R_n(P_{alpha,beta,s}(M,c_0,epsilon_0,t_0),t_0) >= c n^{-2 alpha/(2 alpha+1)} = c rho_n. Thus, in the smooth-covariate regime, the certified lower floor lands on the same exponent as the published benchmark rho_n, but this core does not claim a matching same-class upper bound.
Theorem T-5 verified CausalSmith.Stat.DoseResponseMinimax.frontier_bracket_deficient For every fixed beta>0 and every 0<s<d/4, under the declared same-class assumptions and ass:baseline-submodel-slack, there exists c>0 such that, for all sufficiently large n, R_n(P_{alpha,beta,s}(M,c_0,epsilon_0,t_0),t_0) >= c n^{-2 alpha/(2 alpha+1)}. Moreover rho_n = n^{-2/(1+d/(4s)+1/alpha)} has a strictly smaller exponent than the certified lower floor. The same-class upper endpoint C rho_n is not discharged on the present core.

Citeds (1)

Cited result lem:published-upper-bound-cited verified CausalSmith.Stat.DoseResponseMinimax.publishedUpperBoundCited Cited conditional comparator (Bonvini-Kennedy 2022). Under Assumptions 1-2 and Conditions 1-4 of the source theorem, alpha <= beta, equal covariate smoothness s for the two nuisance functions, projection approximation of order k^(-2s/d), the source's optimal choices of k and h, equal-rate estimation of w and 1/pi, and negligibility of the third-order density-estimation remainder ||v||_g ||q||_g ||p_hat(t,.)-p(t,.)||_infinity, the conditional MSE of the source's specified second-order estimator is eventually at most a constant times rho_n as displayed in Figure 1. This is an estimator-specific conditional risk implication, not an unconditional minimax upper bound over any class and not a same-class upper theorem for def:holder-dose-class.

Auxiliary Lean lemmas (machine-checked) (43)

Definition CausalSmith.Stat.DoseResponseMinimax.MuIsRegression —
Definition CausalSmith.Stat.DoseResponseMinimax.PxIsXDensity —
Definition CausalSmith.Stat.DoseResponseMinimax.PiIsCondTreatmentDensity —
Lemma CausalSmith.Stat.DoseResponseMinimax.minimaxRisk_two_point_lower —
Lemma CausalSmith.Stat.DoseResponseMinimax.measurableSet_cube —
Lemma CausalSmith.Stat.DoseResponseMinimax.restrict_withDensity_ofReal_isProbabilityMeasure —
Lemma CausalSmith.Stat.DoseResponseMinimax.integrable_of_measurable_ae_bounded —
Lemma CausalSmith.Stat.DoseResponseMinimax.measurable_doseObs_tuple —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseBump_holder_gate —
Definition CausalSmith.Stat.DoseResponseMinimax.doseChannelAX —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseAXMeasure_isProbabilityMeasure —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseDataMeasure_eq_AXbind —
Definition CausalSmith.Stat.DoseResponseMinimax.doseContDiffBump —
Definition CausalSmith.Stat.DoseResponseMinimax.doseBump —
Lemma CausalSmith.Stat.DoseResponseMinimax.measurable_twoPointMean_map_doseObs —
Lemma CausalSmith.Stat.DoseResponseMinimax.measurable_twoPointMean_map_doseObs_pair —
Definition CausalSmith.Stat.DoseResponseMinimax.doseXMeasure —
Definition CausalSmith.Stat.DoseResponseMinimax.doseAMeasure —
Lemma CausalSmith.Stat.DoseResponseMinimax.measurable_doseOutcomeKernel —
Definition CausalSmith.Stat.DoseResponseMinimax.doseAXMeasure —
Definition CausalSmith.Stat.DoseResponseMinimax.doseDataMeasure —
Definition CausalSmith.Stat.DoseResponseMinimax.dosePotential —
Definition CausalSmith.Stat.DoseResponseMinimax.doseWitness —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseXMeasure_isProbabilityMeasure —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_kl_single_le —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_kl_nfold_le —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseDataMeasure_isProbabilityMeasure —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseDataMeasure_map_X —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_pxDens —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_consistency —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseDataMeasure_ae_Y_mem_Icc —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_bdd —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_mem_class —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseDataMeasure_map_AX —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseAXMeasure_eq_prod —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseAXMeasure_density —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_piCond —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_muReg —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_ignorability —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_theta —
Lemma CausalSmith.Stat.DoseResponseMinimax.doseWitness_theta_sep —
Lemma CausalSmith.Stat.DoseResponseMinimax.map_bind_map_proj —
Definition CausalSmith.Stat.DoseResponseMinimax.doseWitnessMu —

Symbol realizations (Lean @realizes) (27)

O verified realized by DoseObs, IidSampling
Y verified realized by DoseObs
A verified realized by DoseObs, IidSampling
X verified realized by DoseObs, IidSampling
P(the single-observation data law of the unit O = (Y,A,X); the i.i.d. n-sample O_1,…,O_n is NOT realized here — it is the n-fold product of this law, carried by IidSampling + minimaxRisk, see their verified realized by DoseLaw
tags) verified realized by DoseLaw
P_X verified realized by DoseLaw
mu_P verified realized by DoseLaw
pi_P(carrier [0,1]×[0,1]^d→ℝ; the conditional density A∣X, its declared range [0,∞) enforced by the `0 ≤ π_P` conjunct of `PiIsCondTreatmentDensity` and pinned to the law by that tie's joint factorization) verified realized by DoseLaw
p_{X,P}(carrier [0,1]^d→ℝ; density of P_X, its declared range [0,∞) enforced by the `0 ≤ p_{X,P}` conjunct of `PxHolder` and pinned to the law by the `PxIsXDensity` tie) verified realized by DoseLaw
Y(a) verified realized by DoseLaw
O_1 verified realized by IidSampling, minimaxRisk
... verified realized by IidSampling, minimaxRisk
O_n verified realized by IidSampling, minimaxRisk
p_{X,P}(0 ≤ p_{X,P} on cube; enforces the declared density range [0,∞), so the `ofReal` `PxIsXDensity` tie realizes the actual nonnegative density object) verified realized by PxHolder
pi_P(`0 ≤ π_P` on `[0,1]×cube` enforcing the declared range `[0,∞)`, AND `P.dataMeasure.map (A,X) = (Leb↾[0,1] ⊗ Leb↾[0,1]^d).withDensity (π_P·p_{X,P})`; together the range-clause + joint-factorization tie pin `pi` to the law's genuine nonnegative conditional treatment density) verified realized by PiIsCondTreatmentDensity
pi_P(0 ≤ pi_P on [0,1]×cube; enforces the declared density range [0,∞), so the `ofReal` joint-density tie realizes the actual nonnegative conditional treatment density) verified realized by PiIsCondTreatmentDensity
alpha verified realized by HolderDoseClass, RegimeConstants
beta verified realized by HolderDoseClass, RegimeConstants
s verified realized by HolderDoseClass, RegimeConstants
M verified realized by HolderDoseClass, RegimeConstants
c_0 verified realized by HolderDoseClass, RegimeConstants
t_0 verified realized by HolderDoseClass, RegimeConstants
epsilon_0 verified realized by HolderDoseClass, RegimeConstants
R_n(frontier sequence `n ↦ minimaxRisk M n C t0 : ℝ`, declared space `[0,∞)^ℕ`; pointwise minimax MSE, `def:minimax-risk`) verified realized by HolderDoseClass
evaluation window verified realized by RegimeConstants
rho_n verified realized by publishedHoifRate, publishedHoifRate_nonneg

Proof map 9 statements · 9 edges

A Minimax Lower Bound for Interior Dose-Response Estimation

Abstract

Introduction

Setup and observed-data class

Main results

Discussion, limitations, and extensions

Appendices

auxiliary lemmas and proofs

verification note

Proofs of the main results

References