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Econ-151 · economics · Economics

Minimax causal estimation under bounded variation or nuisance error bounds

LLM status Open unverified

Statement

CI-54 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Use independent binary-baseline observations with $Y\in\{0,1\}$, uniform $X\in[0,1]^d$, consistency, conditional exchangeability, and overlap. Fix $d,L,\varepsilon$. One experiment restricts $e,m_1$ to $\operatorname{BV}_d(L)$, with $0\leq m_1\leq1$. A second experiment supplies known pilot functions $\bar e,\bar m$ satisfying the same pointwise bounds, but makes no derivative or variation assumptions. Its unknown nuisances satisfy \[ \|e-\bar e\|_{L_2(dx)}\leq r_e,\qquad \|m_1-\bar m\|_{L_2(dx)}\leq r_m. \] The pilots are deterministic side information, not estimates fitted to the fresh $n$ observations. Let $r_e=r_{e,n}$ and $r_m=r_{m,n}$ be specified sequences. In both experiments the target is $\psi(P)=\int m_1(x)\,dx$.

Formal question

Determine the squared-error minimax rate in the bounded-variation experiment, including its dependence on fixed dimension. Separately determine \[ \inf_{\widehat\psi}\sup_{\bar e,\bar m} \sup_{\substack{P:\ \|e-\bar e\|_2\leq r_e\\ \|m_1-\bar m\|_2\leq r_m}} \E_P[(\widehat\psi-\psi(P))^2], \] where estimators may use the supplied pilots and the inner supremum retains the causal baseline and overlap restrictions. Give matching bounds across the regimes of $n,r_e,r_m$ and compare what extra information the variation restriction provides.

Scope and status

Structure-agnostic here means an explicitly defined nuisance-error experiment, not a claim of consistent estimation over all measurable nuisance functions. If pilots are random or trained using the estimation sample, their distribution and dependence must instead be included in the experiment. The two classes need not have the same minimax rate.

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 models. 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-151-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-151

Pilot-error bounds and a sharp one-dimensional bounded-variation rate

Status: Unresolved. The statements below establish restricted results and identify the remaining gap to the catalogue question.

Abstract

In the deterministic-pilot experiment, we prove a product-error upper bound with an exact-outcome-pilot collapse and a matching rate when the propensity pilot is exact. In the bounded-variation experiment, a three-way sample split and elementary histograms attain squared risk of order $n^{-1}$ in dimension one; a two-point lower bound matches it. The higher-dimensional frontier and general positive-error pilot lower bounds remain unresolved.

Short target and experiment

Observe independent copies of $(X,W,Y)$ with $X$ uniform on $[0,1]^d$, $W,Y\in\{0,1\}$, and propensity $e(x)\in[\epsilon,1-\epsilon]$ for fixed $0<\epsilon<1/2$. Consistency and exchangeability identify $m(x)={\mathbb{E}}[Y\mid X=x,W=1]$ and $\psi=\int m(x)\,dx$. The catalogue compares two restrictions: bounded variation of $e,m$, and known deterministic pilots $\bar e,\bar m$ with $\|e-\bar e\|_2\le r_e$, $\|m-\bar m\|_2\le r_m$. Pilots obey the same pointwise bounds. Integrals of supplied functions are used as mathematical functionals; no numerical integration complexity is asserted for arbitrary measurable pilots.

The deterministic-pilot experiment

Define \[ \widehat\psi=\int\bar m(x)\,dx+ {1\over n}\sum_{i=1}^n {W_i\over\bar e(X_i)}(Y_i-\bar m(X_i)). \]
Theorem (Uniform risk and zero-radius behavior): The bias is exactly \[ {\mathbb{E}}\widehat\psi-\psi= \int{(e-\bar e)(m-\bar m)\over\bar e}\,dx. \] Consequently the minimax risk $\mathcal R_n(r_e,r_m)$, including the catalogue's supremum over supplied pilots, satisfies \[ \mathcal R_n(r_e,r_m)\le \min\left\{r_m^2,{1\over\epsilon^2 n} +{r_e^2r_m^2\over\epsilon^2}\right\}. \] In particular $r_m=0$ gives exactly zero risk, independently of $r_e$.
Proof: Conditioning on $X$ gives expected correction $e(m-\bar m)/\bar e$. Subtracting $m$ after adding $\bar m$ yields the displayed product. Cauchy--Schwarz bounds its magnitude by $r_er_m/\epsilon$. Each summand has absolute value at most $1/\epsilon$, so the variance of its average is at most $1/(\epsilon^2n)$. Squared bias plus variance proves the second candidate bound. The deterministic estimator $\int\bar m$ has error at most $\|m-\bar m\|_2\le r_m$, since the design domain has volume one. Choose whichever guaranteed bound is smaller, using the supplied radii. If $r_m=0$, equality almost everywhere makes the target exactly the known pilot integral. ∎
Theorem (A lower bound and a solved pilot slice): For all radii and $n\ge1$, with constants depending only on $\epsilon$, \[ \mathcal R_n(r_e,r_m)\ge c\min\{r_m^2,n^{-1}\}. \] When $r_e=0$, this lower bound matches the upper bound up to constants; the minimax squared risk is therefore of order $\min\{r_m^2,n^{-1}\}$.
Proof: Choose supplied pilots $\bar e=\bar m=1/2$, true $e=1/2$, and outcome regressions $m_\pm=1/2\pm\delta$, where $\delta=c_0\min\{r_m,n^{-1/2}\}$ for a sufficiently small fixed $c_0>0$. Use a fair treatment coin independent of Bernoulli potential outcomes, and make the control outcome distribution the same under both models. Both laws belong to every specified $r_e$ class and the required $r_m$ class. Their target difference is $2\delta$.

For Bernoulli probabilities in $[1/4,3/4]$, the one-observation divergence is at most a constant times $\delta^2$; the treatment coin only reduces it. Thus the $n$-sample divergence is at most $C n\delta^2$, made at most $1/2$ by choosing $c_0$ small. Pinsker gives total variation at most $1/2$. Any estimator closer to the wrong target incurs error at least $\delta$; testing the two equally likely models therefore has average squared error at least $\delta^2(1-\operatorname{TV})/2\ge\delta^2/4$. This proves the lower bound. Substituting $r_e=0$ into the upper theorem proves the matching assertion. ∎

The product bias can be exact: for constant pilots $\bar e=\bar m=1/2$ and small admissible constants $e-\bar e=a$, $m-\bar m=b$, it equals $2ab$. This demonstrates sharpness of a bias bound for this estimator. It is not a minimax lower bound of order $a^2b^2$, because fresh observations can estimate these constant nuisances.

One-dimensional bounded variation

Now take $d=1$ and assume $e,m$ have total variation at most $L$, with the pointwise bounds already stated. For the catalogue's $\operatorname{BV}_1(L)$ ball, this total-variation condition follows from its stronger norm bound. Divide $n$ observations into three independent blocks of size $N=\lfloor n/3\rfloor$. Use equal bins of length $h=1/q$, with $q$ of order $\sqrt N$.
Lemma (Histogram nuisance risks): There are estimates from the first block for $e$ and from the second block for $m$, clipped to their pointwise ranges, such that \[ {\mathbb{E}}\|\widehat e-e\|_2^2+{\mathbb{E}}\|\widehat m-m\|_2^2 \le C\{h+(Nh)^{-1}\}. \] The two estimated functions are independent.
Proof: For any one-dimensional function $f$ of variation at most $V$, its binwise mean $f_h$ satisfies \[ \|f-f_h\|_2^2\le h\sum_B\operatorname{osc}(f;B)^2 \le h V^2. \] Choose a bounded-variation representative; endpoints have no effect on the integral. The sum of within-bin variations is at most total variation, proving the final inequality. If a bounded response $Z\in[0,1]$ has conditional mean $f$, estimate its bin mean by $(Nh)^{-1}\sum Z_i{\mathbf{1}}\{X_i\in B\}$. This has the correct expectation and integrated variance at most $1/(Nh)$.

Apply this to $Z=W$ on block one and clip to $[\epsilon,1-\epsilon]$ to estimate $e$. On block two estimate both $e$ and $g(x)={\mathbb{E}}[WY\mid X=x]=e(x)m(x)$ by the same construction. The product variation inequality gives $\operatorname{TV}(g)\le\operatorname{TV}(e)+\operatorname{TV}(m)\le2L$. Clip the block-two propensity estimate to $[\epsilon,1-\epsilon]$, call it $\widetilde e$, and put $\widehat m=\operatorname{clip}_{[0,1]}(\widetilde g/\widetilde e)$. Pointwise, \[ |\widehat m-m|\le {1\over\epsilon} (|\widetilde g-g|+|\widetilde e-e|), \] since $g=em$ and $0\le m\le1$. Squaring and integrating proves the bound. All operations for $\widehat e$ use the first block and all for $\widehat m$ the second, proving independence. ∎

Use the third block in the previous correction estimator, replacing the pilots by $\widehat e,\widehat m$. The integral of the histogram $\widehat m$ is a finite sum and can be computed exactly.
Theorem (Sharp $d=1$ rate): For fixed $\epsilon$ and a nondegenerate bounded-variation radius, for example the catalogue norm radius $L\ge1$, the one-dimensional experiment has minimax squared risk of order $n^{-1}$. The split-sample estimator just described attains the upper bound uniformly.
Proof: Conditional on both training blocks, the bias identity from the first theorem still holds, and validation variance is at most $1/(\epsilon^2N)$. The conditional squared bias is at most $\epsilon^{-2}\|\widehat e-e\|_2^2\|\widehat m-m\|_2^2$. Independence of the training blocks makes the expectation of this product the product of their expectations. Choose $h$ of order $N^{-1/2}$ in the lemma: each expected squared nuisance error is $O(N^{-1/2})$, so the expected squared bias is $O(N^{-1})$. Adding validation variance gives $C/n$ for $n\ge6$; enlarging the constant handles smaller samples with a bounded estimator.

For the lower bound, the constant-regression two-point family from the previous theorem, with $\delta=c_0n^{-1/2}$, lies in the bounded-variation ball when $L\ge1$. Its total variation is zero and its $L^1$ norms are at most one. The same testing argument gives $c/n$. ∎

This $n^{-1}$ squared-risk result (root-$n$ estimation) is not a semiparametric efficiency or asymptotic normality theorem. It uses the known uniform covariate distribution and two independent nuisance fits; multiplying expectation bounds without that independence would require further moment control.

Unresolved regimes and source

For $d>1$, the one-dimensional oscillation argument is not a dimension-free bounded-variation approximation theorem. For $r_e>0$, the displayed pilot lower and upper bounds can differ, and no matching product-error lower bound has been proved here. Those gaps prevent a full resolution of the catalogue's two minimax experiments.

References

  • [CF] 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. Source of the new-models agenda. The elementary restricted bounds above are fully derived and are not presented as a literature-wide novelty claim.

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