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Econ-56-A2 · revision r1 · on Econ-56

GPT 6 Astra Ultra attempt on Econ-56, version 2

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Contents
  1. Minimax estimation of longitudinal intervention effects: Direct nested regression and an attained five-index parametric region
  2. Target
  3. Exact two-stage bias
  4. An exact result with supplied assignment probabilities
  5. A direct nested regression with explicit attainable rates
  6. Unresolved minimax interactions
  7. Completion Estimate
  8. References

Minimax estimation of longitudinal intervention effects: Direct nested regression and an attained five-index parametric region

Problem ID: C14-55. Status: Unresolved; rigorous restricted results.

Target

For a fixed two-stage regime $a=(a_0,a_1)$, write $e_0(x)=P(A_0=a_0\mid X=x)$, $e_1(x,l)=P(A_1=a_1\mid X=x,A_0=a_0,L=l)$, $g(l\mid x)=p(l\mid x,A_0=a_0)$, and $m(x,l)=\mathbb E[Y\mid x,a_0,l,a_1]$. Then \[ Q(x)=\int m(x,l)g(l\mid x)dl,\qquad \psi_a=\int Q(x)p(x)dx. \] The canonical question seeks the squared-error minimax rate under distinct Hölder restrictions on all five nuisance functions. The agenda is [agenda, Section 3.5.3]. Longitudinal higher-order methodology already exists, including [robins]; this note supplies an exact first-order remainder, a constructive direct-regression upper bound with a matching parametric subregion, and a sharp known-assignment experiment. The full longitudinal smoothness phase diagram remains unresolved.

Exact two-stage bias

Train pilots $h_0,h_1,\widehat m,\widehat Q$ independently of an evaluation block of size $N$, with $h_t\in[\varepsilon,1-\varepsilon]$ and outcome pilots in $[0,1]$. Put $I_t=\mathbf1\{A_t=a_t\}$ and average \[ S=\widehat Q(X)+\frac{I_0}{h_0(X)} \{\widehat m(X,L)-\widehat Q(X)\} +\frac{I_0I_1}{h_0(X)h_1(X,L)}\{Y-\widehat m(X,L)\}. \tag{1} \] Let $\mu(dx,dl)=p(x)g(l\mid x)dx\,dl$.
Theorem: Conditionally on the pilots, the bias of (1) is \[ B=\int p\left(1-\frac{e_0}{h_0}\right)(\widehat Q-Q)dx +\int\frac{e_0}{h_0}\left(1-\frac{e_1}{h_1}\right) (\widehat m-m)d\mu. \tag{2} \] Consequently its projected mean has risk bounded by \[ \frac{C_\varepsilon}{N}+C_\varepsilon\mathbb E\left[ \|h_0-e_0\|_{L^2(p)}^2\|\widehat Q-Q\|_{L^2(p)}^2 +\|h_1-e_1\|_{L^2(\mu)}^2\|\widehat m-m\|_{L^2(\mu)}^2\right]. \tag{3} \]
Proof: Conditional on $X=x$, the expectation of the second and third terms in (1) equals \[ \frac{e_0}{h_0}\int g\left\{\widehat m-\widehat Q +\frac{e_1}{h_1}(m-\widehat m)\right\}dl =\frac{e_0}{h_0}\left\{Q-\widehat Q+ \int g(1-e_1/h_1)(\widehat m-m)dl\right\}. \] Adding $\widehat Q$ proves (2). Cauchy--Schwarz and the clipped denominator bounds give the two products in (3), after $(x+y)^2\le2x^2+2y^2$. The score is bounded by $1+\varepsilon^{-1}+\varepsilon^{-2}$, so its conditional variance divided by $N$ proves the remaining term. ∎
Proposition: If $g\le G$, $\widehat g$ is a nonnegative conditional density, and $\widehat Q(x)=\int\widehat m(x,l)\widehat g(l\mid x)dl$, then \[ \|\widehat Q-Q\|_{L^2(p)} \le \|\widehat m-m\|_{L^2(\mu)} +\|\widehat g-g\|_{L^2(p(x)dx\,dl)}. \tag{4} \] In particular, suppose the evaluation size satisfies $N\asymp n$, and all four pilot errors have fourth moments of orders $n^{-4\gamma_0},n^{-4\gamma_1},n^{-4\gamma_m},n^{-4\gamma_g}$ in these respective norms. Then (3) is at most \[ C\{n^{-1}+n^{-2(\gamma_0+\min(\gamma_m,\gamma_g))} +n^{-2(\gamma_1+\gamma_m)}\}. \tag{5} \]
Proof: Write $\widehat Q-Q=\int(\widehat m-m)g+ \int\widehat m(\widehat g-g)$. Jensen bounds the first term's norm by the first term on the right of (4). Cauchy--Schwarz in $l$, whose unit cube has volume one, and $0\le\widehat m\le1$ bound the second term. Minkowski proves (4). For (5), apply Cauchy--Schwarz to the expectations of products in (3), then (4) and $(x+y)^4\le8(x^4+y^4)$. ∎
Thus an explicitly sufficient parametric region for any pilots satisfying these moment bounds is $\gamma_0+\min(\gamma_m,\gamma_g)\ge1/2$ and $\gamma_1+\gamma_m\ge1/2$. This is a conditional statement about pilot performance, not an unproved assertion that every indicated exponent is attainable for arbitrary joint Hölder indices. Formula (2) also shows that density and transition errors do not necessarily enter as independent additive first-order errors. In particular, a sufficiently good direct $Q$ estimator may be preferable to separately estimating $g$.

An exact result with supplied assignment probabilities

Consider $T$ treatment decisions, each with the probability of the requested action at least $\varepsilon$. Histories and their transitions may otherwise be arbitrary. Let $p_t$ denote those supplied, history-dependent probabilities, $Y\in[0,1]$, and retain sequential exchangeability and consistency.
Theorem: The minimax risk over this known-assignment experiment is at most \[ \min\{1,\,(n\varepsilon^T)^{-1}\}. \] Over the subexperiment with independent assignments of probability $\varepsilon$, ancillary uniform continuous histories, and an unknown constant Bernoulli outcome mean only on the requested treatment path, it is at least $c\min\{1,(n\varepsilon^T)^{-1}\}$ for a universal $c>0$. All continuous-coordinate transition and outcome functions in this subexperiment are smooth of every order.
Proof: Let $R=\prod_{t=1}^T\mathbf1\{A_t=a_t\}/p_t(H_t)$. Iterated conditional expectations, or cancellation of the assignment factors in the joint density, give $\mathbb E[RY]=\psi_a$ and $\mathbb E R=1$. Since $R\le\varepsilon^{-T}$, $\mathbb E(RY)^2\le\mathbb E R^2\le\varepsilon^{-T}$. The projected empirical mean therefore has risk at most $(n\varepsilon^T)^{-1}$; the constant estimator gives the other bound. For the lower bound let $q=\varepsilon^T$, set all off-path outcome means to $1/2$, and set the on-path mean to $1/2\pm\Delta$. The one-observation KL is at most $Cq\Delta^2$ for $0<\Delta\le1/4$. Choose $\Delta=b\min\{1,(nq)^{-1/2}\}$ with $b$ small. Then sample total variation is at most $1/2$ and the two targets differ by $2\Delta$. Testing by the nearer target gives maximal squared risk at least $\Delta^2/4$. Uniform ancillary histories do not change this likelihood calculation or any smoothness constraint with sufficient fixed radius. ∎

A direct nested regression with explicit attainable rates

This section constructs all pilots rather than assuming their moment rates. It uses the canonical five-function model and the fixed two-stage horizon. Let $D=d_0+d_1$, and set \[ t_0=\min(s_0,1),\quad t_1=\min(s_1,1),\quad t_m=\min(s_m,1),\quad t_Q=\min(s_m,s_g,1), \] \[ \gamma_0=\frac{t_0}{2t_0+d_0},\quad \gamma_1=\frac{t_1}{2t_1+D},\quad \gamma_m=\frac{t_m}{2t_m+D},\quad \gamma_Q=\frac{t_Q}{2t_Q+d_0}. \tag{6} \] All these indices are supplied for this minimax upper bound. We make no adaptation claim here. The construction estimates $Q$ directly from a pseudo-outcome and never estimates $g$ or $p$.
Lemma (Integrated regression and contamination): The integrated regression $Q$ has a Hölder modulus of order $t_Q$, with a constant depending only on the fixed dimensions and radii. Let $\widetilde m:[0,1]^D\to[0,1]$ be any random function independent of a new block of $N$ trajectories. On that block regress the response $\widetilde m(X,L)$ among observations with $A_0=a_0$ on $X$ by a histogram of side length $h$, using $1/2$ in empty cells. Call the result $\widetilde Q$. Then, conditionally on $\widetilde m$, \[ \mathbb E[\|\widetilde Q-Q\|_{L^2(p)}^2\mid\widetilde m] \le C\left\{h^{2t_Q}+\frac{h^{-d_0}}{N+1} +\|\widetilde m-m\|_{L^2(\mu)}^2\right\}. \tag{7} \] No smoothness of the fitted function $\widetilde m$ is required.
Proof: For two covariate points, add and subtract $m(x',l)g(l\mid x)$: \[ Q(x)-Q(x')=\int[m(x,l)-m(x',l)]g(l\mid x)dl +\int m(x',l)[g(l\mid x)-g(l\mid x')]dl. \] The first term is bounded by a constant times $\|x-x'\|^{\min(s_m,1)}$, since $g$ integrates to one. The second is bounded by a constant times $\|x-x'\|^{\min(s_g,1)}$, since $0\le m\le1$ and the $l$-cube has volume one. Indices above one imply Lipschitz moduli through bounded first derivatives. The bounded support absorbs the constants needed to replace both exponents by $t_Q$.

For an $X$-cell $A$, write $\pi_A=\int_Ap$ and $\rho_A=\int_A e_0p\ge\varepsilon\pi_A$. The true selected mean of $Q$ is $q_A=\rho_A^{-1}\int_A e_0p Q$ and differs from every $Q(x)$ in that cell by at most $Ch^{t_Q}$. Conditional on $\widetilde m$, the selected pseudo-response mean in the cell is $q_A+r_A$, where \[ r_A=\frac1{\rho_A}\int_A e_0(x)p(x) \int [\widetilde m(x,l)-m(x,l)]g(l\mid x)\,dl\,dx. \] Jensen's inequality for the selected conditional law gives \[ \sum_A\pi_A r_A^2 \le\sum_A\frac{\pi_A}{\rho_A}\int_Ae_0p \int(\widetilde m-m)^2g\,dl\,dx \le\varepsilon^{-1}\|\widetilde m-m\|_{L^2(\mu)}^2. \tag{8} \] On a positive count $j$, the empirical pseudo-response mean is unbiased for $q_A+r_A$ and has conditional variance at most $1/(4j)$. The binomial reciprocal identity and empty-cell bound used earlier give an integrated sampling contribution at most $C h^{-d_0}/(N+1)$. Using $(u+v)^2\le2u^2+2v^2$ for approximation and contamination, then (8), proves (7). The argument conditioned on the whole random function first; it therefore does not assume that a fitted histogram is smooth or that its evaluation errors are independent across cells. ∎

Theorem (Constructive five-index upper bound): There is an explicitly tuned, five-block estimator with \[ \sup_{P\in\mathcal P(s_p,s_0,s_g,s_1,s_m)} \mathbb E_P(\widehat\psi_a-\psi_a)^2 \le C\left\{n^{-1}+n^{-2(\gamma_0+\gamma_Q)} +n^{-2(\gamma_0+\gamma_m)}+n^{-2(\gamma_1+\gamma_m)}\right\}. \tag{9} \] Its constant depends only on the fixed dimensions, indices, radii, density bounds and overlap. Thus \[ \gamma_0+\min(\gamma_Q,\gamma_m)\ge\tfrac12,\qquad \gamma_1+\gamma_m\ge\tfrac12 \tag{10} \] is an attained parametric-risk region. If the class contains a fixed nondegenerate constant Bernoulli outcome subfamily with uniform histories and constant assignments, its minimax risk is $\Theta(n^{-1})$ throughout (10).
Proof: For sufficiently large $n$, use five disjoint blocks of size $N=\lfloor n/5\rfloor$. On block 1 fit $h_0$ by regressing $I_0=\mathbf1\{A_0=a_0\}$ on $X$. On block 2 fit $h_1$ by regressing $I_1=\mathbf1\{A_1=a_1\}$ on $(X,L)$ among $I_0=1$. On block 3 fit $\widehat m$ by regressing $Y$ on $(X,L)$ among $I_0I_1=1$. Use equal-cube histograms with side lengths within a fixed factor of $N^{-1/(2t_0+d_0)}$, $N^{-1/(2t_1+D)}$, and $N^{-1/(2t_m+D)}$, respectively. Clip the propensity fits to $[\varepsilon,1-\varepsilon]$ and the outcome fit to $[0,1]$.

Here is the elementary risk calculation for these pilots. For each cell the regression oscillation is at most a constant times its side length to the relevant $t$ power. The positive-count response mean is unbiased for its selected cell mean and has variance at most $1/(4j)$. For the $h_1$ fit the selected cell probability is $\int_{\text{cell}}e_0\,d\mu\ge\varepsilon\mu(\text{cell})$; for the $\widehat m$ fit it is $\int_{\text{cell}}e_0e_1\,d\mu\ge\varepsilon^2\mu(\text{cell})$. For $h_0$ it is $P_X(\text{cell})$. Multiplying the binomial reciprocal and empty-cell bounds by the respective target cell masses and summing gives risk at most $C\{h^{2t}+h^{-d'}/(N+1)\}$ in dimension $d'$. This argument uses the intervention design measure $\mu=p g$ for the last two errors; it does not silently replace it by the observed marginal law of $(X,L)$. The resulting bounds are \[ \mathbb E\|h_0-e_0\|_{L^2(p)}^2\le Cn^{-2\gamma_0},\quad \mathbb E\|h_1-e_1\|_{L^2(\mu)}^2\le Cn^{-2\gamma_1},\quad \mathbb E\|\widehat m-m\|_{L^2(\mu)}^2\le Cn^{-2\gamma_m}. \tag{11} \]

On block 4 construct $\widehat Q$ from pseudo-response $\widehat m(X,L)$ as in (7), using side length $h_Q\asymp N^{-1/(2t_Q+d_0)}$. Independence of blocks 3 and 4, followed by (7) and (11), gives \[ \mathbb E\|\widehat Q-Q\|_{L^2(p)}^2 \le C\{n^{-2\gamma_Q}+n^{-2\gamma_m}\}. \tag{12} \] The fit already lies in $[0,1]$. On block 5 average the longitudinal score (1) and project its average to $[0,1]$.

The exact remainder (2) and bounded-score variance give (3). The function $h_0$ uses only block 1, whereas $\widehat Q$ uses only blocks 3 and 4. Thus their squared errors in the first product in (3) are independent. Similarly $h_1$ uses only block 2 and $\widehat m$ only block 3, so the second product also factors in expectation. Independence of $\widehat Q$ and $\widehat m$ is neither true nor needed. Substitution of (11)--(12) into (3) proves (9) using only second-moment pilot bounds, without the fourth-moment premise of (5). Small sample sizes are handled by the constant estimator and a larger constant. Condition (10) makes all displayed terms $O(n^{-1})$.

For the matching lower bound in that region, let histories be independent uniform variables, assignments independent Bernoulli$(1/2)$, and $Y\mid X,A_0,L,A_1\sim\operatorname{Ber}(\theta)$ with $\theta=1/2\pm b/\sqrt n$. The target is $\theta$ and the outcome law is constant in every continuous coordinate, hence lies in every stated smoothness ball under the explicit subfamily assumption. Product KL is $O(b^2)$. Choose fixed $b>0$ so total variation is at most $1/2$; the nearer-target test yields a lower bound $c/n$ for squared-error risk. Together with (9) this proves the restricted sharp rate. ∎

For a concrete improvement, take $d_0=d_1=1$, $s_0=s_1=s_m=1$ and $s_g=2/5$, with any positive $s_p$. Then \[ \gamma_0=1/3,\qquad \gamma_1=\gamma_m=1/4,\qquad \gamma_Q=2/9, \] so (10) holds. Separate histogram estimation of the transition density would instead yield \[ \gamma_g=\frac{2/5}{2(2/5)+2}=\frac17,\qquad \gamma_0+\min(\gamma_m,\gamma_g)=\frac{10}{21}<\frac12, \] which does not satisfy the earlier sufficient condition. The comparison concerns these proved first-order constructions; it is not a lower bound against every estimator that explicitly estimates $g$.

Unresolved minimax interactions

For fixed $T$, known assignment yields a parametric experiment, while for growing $T$ even completely smooth submodels require $n\varepsilon^T$ to diverge. The five-block construction now attains the displayed sufficient smoothness region with unknown propensities, without separately estimating the transition density. Its sufficient inequalities are not claimed to be necessary. Higher-order corrections, matching lower bounds in the slow regions, the full sharp five-index phase diagram and its fixed-multistage extension remain unresolved in this attempt.

Completion Estimate

35%

This is a heuristic estimate for the full catalogue target. The direct nested-regression theorem supplies actual pilot constructions, removes the earlier fourth-moment oracle premise for this bound and strictly enlarges a proved parametric region. A matching general lower bound across the slower five-index regions and the fixed-multistage characterization are still missing.

References

  • [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025), arXiv:2508.17099v1. https://arxiv.org/html/2508.17099v1.
  • [robins] J. Robins, L. Li, E. Tchetgen Tchetgen and A. van der Vaart. Technical Report: Higher Order Influence Functions and Minimax Estimation of Nonlinear Functionals (2016). https://arxiv.org/abs/1601.05820.