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Econ-103 · economics · Economics

Minimax rates for dose--response curves

LLM status Open unverified

Statement

CI-49 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Observe independent $O=(X,A,Y)$ with $X\in[0,1]^d$, continuous dose $A\in[0,1]$, and $|Y|\leq M$. Assume a jointly measurable structural representation $Y(a)=F(a,X,U)$ and $Y=F(A,X,U)$, with $U\perp A\mid X$, and conditional dose density $c\leq g(a\mid x)\leq C$. Require the causal conditional response $a\mapsto\E[F(a,x,U)\mid X=x]$ to be continuous for almost every $x$. The structural independence concerns the full response mechanism, ruling out exceptional-diagonal counterexamples that pointwise ignorability does not exclude. Let $\mu(a,x)$ denote the smooth version of the observed conditional mean. Fix positive indices and require \[ \mu\in\mathcal H_{d+1}^{s_\mu}(L),\quad g\in\mathcal H_{d+1}^{s_g}(L),\quad p\in\mathcal H_d^{s_p}(L),\quad c\leq p\leq C. \] For fixed $\eta\in(0,1/2)$, take $I=[\eta,1-\eta]$. The causal continuity assumption identifies $\theta(a)=\E[Y(a)]=\int\mu(a,x)p(x)\,dx$ at every $a\in I$.

Formal question

Find matching asymptotic bounds, at fixed $d$ and smoothness indices, for each of \[ \inf_{\widehat\theta}\sup_{P\in\mathcal P} \E_P\int_I(\widehat\theta(a)-\theta(a))^2\,da, \qquad \inf_{\widehat\theta}\sup_{P\in\mathcal P} \E_P\|\widehat\theta-\theta\|_{\infty,I}^2. \] Characterize the effects of smoothing in dose, covariate dimension, unknown dose density, and unknown covariate density. Supply rate-attaining estimators and compare oracle experiments with supplied nuisance densities; state any logarithmic penalty under supremum loss.

Scope and status

A continuously indexed response curve is not a regular scalar average effect, so a scalar root-$n$ benchmark cannot simply be transferred to it. The interior domain, causal continuity, and losses make the estimation problem unambiguous.

References for the source of the problem

Carlos Cinelli, Avi Feller, Guido Imbens, Edward Kennedy, Sara Magliacane, and Jose Zubizarreta, Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1, 23 August 2025, Section 3.5.3, New parameters. https://arxiv.org/html/2508.17099v1#S3.SS5.SSS3

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-103-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-103

Oracle dose-response rates and an exact nuisance remainder

Problem ID: C21-101. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

The restricted question

The target is $\theta(a)=\int\mu(a,x)p(x)dx$ on an interior dose interval $I=[\eta,1-\eta]$. Observations are independent $(X,A,Y)$, $|Y|\le M$, with a conditional dose density $c\le g\le C$. The catalogue's structural independence and continuity assumptions identify this version of the curve at every interior dose. The full question concerns arbitrary positive smoothness indices and unknown $g,p$.

We establish both oracle minimax rates for $0<s=s_\mu\le1$ and derive a conditional risk inequality for an estimator using independently trained nuisance estimates. These results distinguish the one-dimensional smoothing cost from a remaining nuisance product. They do not settle the general unknown-nuisance minimax frontier. For lower bounds the class constants are assumed to contain a fixed neighborhood of the uniform design and a constant Bernoulli regression; otherwise a ball could be degenerate.

Oracle estimator and upper bounds

Partition $I$ into $J$ intervals of equal width $h=|I|/J$. For $a\in B_j$, when $g$ is supplied, define \[ \widehat\theta(a)=\frac1{nh}\sum_{i=1}^n \frac{Y_i{\mathbf 1}\{A_i\in B_j\}}{g(A_i\mid X_i)}. \] Only the dose density is supplied; the covariate density is still unknown.
Theorem (Oracle minimax rates for order at most one): For fixed $0<s\le1$, the oracle integrated squared minimax risk is of order $n^{-2s/(2s+1)}$. Its squared supremum minimax risk on $I$ is of order $(\log n/n)^{2s/(2s+1)}$. Constants may depend on the fixed model bounds and $I$, but not $n$.
Proof[Upper bounds]: The expectation in bin $j$ is $\bar\theta_j=h^{-1}\int_{B_j}\theta(t)dt$, since the inverse conditional dose density cancels $g$. Marginalization preserves the $s$-Hölder bound in dose, so $|\bar\theta_j-\theta(a)|\le Lh^s$. The unscaled summand has absolute value at most $M/c$ and second moment at most $M^2h/c$. Hence each bin estimate has variance at most $M^2/(cnh)$. Integration gives \[ {\mathbb{E}}\|\widehat\theta-\theta\|_{2,I}^2 \le C_1\{h^{2s}+(nh)^{-1}\}. \] For the supremum, a bounded-summand exponential-moment argument yields \[ {\mathbb{P}}\left\{\max_j|\widehat\theta_j-\bar\theta_j| >C_2\left(\sqrt{u/(nh)}+u/(nh)\right)\right\} \le2Je^{-u}. \tag{1} \] One direct derivation bounds each centered summand's $k$th absolute moment by its variance times its absolute bound to power $k-2$, sums the exponential series, and applies exponential Markov; union bounding over bins gives (1). Taking $u=\log(2J)+v$ and integrating $e^{-v}$ proves \[ {\mathbb{E}}\|\widehat\theta-\theta\|_{\infty,I}^2 \le C_3\left\{h^{2s}+\frac{\log(2J)}{nh} +\frac{\log^2(2J)}{n^2h^2}\right\}. \] Choose $h$ of order $n^{-1/(2s+1)}$ or $(\log n/n)^{1/(2s+1)}$, respectively, rounding $J$ to an integer. These prove the upper bounds. ∎

Self-contained lower-bound constructions

After a fixed rescaling of outcome units if $M<1$, take $X,A$ independent uniform and $Y\mid A=a$ Bernoulli with probability $\theta(a)\in[1/4,3/4]$. Set $Y(a)={\mathbf 1}\{U\le\theta(a)\}$ for one independent uniform $U$. Multiplying these outcomes and their target means by any fixed positive constant only changes the risk constants. This gives the required structural independence and continuous mean; the conditional mean is independent of $x$, and supplied densities $g=p=1$ carry no information about $\theta$.

Fix a nonzero smooth bump $\phi$ supported in $(0,1)$, with maximum one. Put disjoint translated bumps $\phi_j(a)=\phi((a-a_j)/h)$ in an interior subinterval, with $J\asymp1/h$. At amplitude $\delta=\epsilon h^s$, all curves below satisfy the Hölder restrictions for small fixed $\epsilon$. For two Bernoulli probabilities in $[1/4,3/4]$, using $\log z\le z-1$ gives \[ {\operatorname{KL}}(\operatorname{Ber}(u),\operatorname{Ber}(v)) \le\frac{(u-v)^2}{v(1-v)}\le\tfrac{16}{3}(u-v)^2. \tag{2} \] Thus sample KL divergence is at most a constant times $n$ times squared $L^2$ distance of the regression curves.

Lemma (Integrated lower bound): The above submodel has integrated minimax risk at least a constant times $n^{-2s/(2s+1)}$.
Proof: Use $\theta_\omega=1/2+\delta\sum_{j=1}^J\omega_j\phi_j$, with independent signs $\omega_j\in\{-1,1\}$. Decode sign $j$ from the sign of $\int_{B_j}(\widehat\theta-1/2)\phi_j$. If the sign is wrong, Cauchy--Schwarz implies squared estimation error on this bin is at least $\delta^2h\int\phi^2$. Adjacent sign experiments have sample KL at most $C n\delta^2h$ by (2). Choose $h\asymp n^{-1/(2s+1)}$ and $\epsilon$ small, so their total variation is at most $1/2$. Here the inequality ${\operatorname{TV}}\le\sqrt{{\operatorname{KL}}/2}$ follows, for example, by mapping a density pair to the set where the first is larger, applying the log-sum inequality, and using binary KL $\ge2(u-v)^2$; the latter follows by differentiating twice in $u$.

For each fixed configuration of other signs, the sum of the two testing errors is at least $1-{\operatorname{TV}}\ge1/2$. Averaging all signs gives error probability at least $1/4$ per bit. Sum the bin losses to get $cJ\delta^2h\asymp\delta^2$, as required. ∎

Lemma (Supremum lower bound): The same submodel has squared supremum minimax risk at least a constant times $(\log n/n)^{2s/(2s+1)}$.
Proof: Use $J$ alternatives $\theta_j=1/2+\delta\phi_j$ and reference $\theta_0=1/2$. Distinct alternatives are separated by $\delta$ in supremum norm. Choose $h\asymp(\log n/n)^{1/(2s+1)}$ and small $\epsilon$, so (2) gives ${\operatorname{KL}}(P_j^{\otimes n},P_0^{\otimes n})\le(\log J)/8$ for all large $n$.

For clarity the testing bound used here is as follows. Let $V$ be uniform on $\{1,\ldots,J\}$, with data law $P_V$. Its mutual information with the data is at most $J^{-1}\sum_j{\operatorname{KL}}(P_j,Q)$ for any reference law $Q$: subtracting the actual mixture-law KL leaves the nonnegative KL from the mixture to $Q$. For any decoder with error probability $p_e$, the conditional entropy of $V$ is at most $\log2+p_e\log J$. Indeed, condition additionally on whether the decoder is correct; the binary indicator has entropy at most $\log2$, and on error at most $J-1$ indices remain. Since $H(V)=\log J$, these facts imply \[ p_e\ge1-\frac{I(V;\text{data})+\log2}{\log J}\ge\tfrac12 \] for all large $J$. Select the nearest alternative to any function estimator; a decoding error forces supremum estimation error at least $\delta/2$. Its average squared risk is consequently at least $\delta^2/8$. This proves the claim and finishes the lower bounds for Theorem 2.1. ∎

The exact remainder when the density is estimated

Use a training sample independent of a validation sample of size $n$. Condition on trained functions $\widetilde\mu,\widetilde g$ satisfying $|\widetilde\mu|\le M$, $\widetilde g\ge c/2$. For bin $j$, set \[ Z_j(O)=\frac1h\int_{B_j}\widetilde\mu(t,X)dt+ \frac{{\mathbf 1}\{A\in B_j\}}{h\widetilde g(A\mid X)} [Y-\widetilde\mu(A,X)],\qquad \widetilde\theta_j=\frac1n\sum_i Z_j(O_i). \] Let $a_0=\|\widetilde\mu-\mu\|_\infty$ and $b_0=\|\widetilde g-g\|_\infty$ on $[0,1]^{d+1}$.
Proposition (Conditional product remainder): Conditionally on training, \[ {\mathbb{E}} Z_j-\bar\theta_j =\frac1h\int_{B_j}\int \frac{(\widetilde\mu-\mu)(\widetilde g-g)}{\widetilde g}(t,x) p(x)\,dx\,dt. \tag{3} \] The integrated conditional risk is at most \[ C\{h^{2s}+(nh)^{-1}+a_0^2b_0^2\}, \] and the supremum conditional risk is at most \[ C\{h^{2s}+\log(2J)/(nh)+\log^2(2J)/(n^2h^2)+a_0^2b_0^2\}. \]
Proof: Condition on $(X,A)$ in the residual term and integrate using its true conditional density $g$. The sum of its mean and the first term is $\widetilde\mu+(g/\widetilde g)(\mu-\widetilde\mu)$ inside the bin integral. Subtract $\mu$ and factor to obtain (3), bounded by $2a_0b_0/c$. The first term of $Z_j$ is bounded by $M$, while its residual term is bounded by $4M/(ch)$ and has second moment at most a constant times $1/h$ because $g\le C$. The same variance and maximum inequalities as before apply, with changed constants. Add bin bias and (3) using $(u+v+w)^2\le3(u^2+v^2+w^2)$. ∎

No estimator of $p$ appears: fresh covariates perform marginalization. This does not make $s_p$ irrelevant to the full minimax problem, because nuisance learning and possible higher-order corrections still depend on the experiment. The proposition supplies a sufficient condition: if the expected squared nuisance product is no larger than the desired oracle risk, the corresponding oracle rate follows after accounting for sample splitting. It is not a necessary condition and is not a lower bound for unknown nuisances. Smoothness above one, weaker nuisance norms, the low-smoothness frontier, and sharp joint effects of $s_\mu,s_g,s_p,d$ remain unresolved.

References

  • [source] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta (2025), Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1, Section 3.5.3. https://arxiv.org/html/2508.17099v1#S3.SS5.SSS3. The survey supplies the agenda; histogram, testing, and remainder arguments are written out here, with no claim that these classical techniques are new.

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