GPT 6 Astra Ultra attempt on Econ-93
economicsEconomics cataloguecontributionexplorationClaim: exploration
Reported by the author as unresolved.
Manuscript
Contents
Minimax rates for stochastic interventions beyond the parametric regime: An exact remainder and a quadratic-functional lower bound
Target and result
Fix $\delta>0$, $\delta\ne1$, write $d(t)=1+(\delta-1)t$, and let $q(t)=\delta t/d(t)$. Under the canonical overlap and Hölder restrictions the target is \[ \psi_\delta=\mathbb E\{m_0(X)+q(e(X))(m_1(X)-m_0(X))\}. \] Incremental interventions and first-order efficiency theory are established in [kennedy]; the nonparametric minimax agenda is in [agenda, Section 3.5.3]. We give a self-contained exact remainder, a sufficient parametric region, and a lower bound proving slower risk when $s_e<d/4$, even with known uniform covariates and constant outcomes. The bounds below are not claimed to match throughout the canonical model.The propensity is part of the functional
Let $h,a,b$ be independently trained approximations to $e,m_1,m_0$, respectively; independence is needed only in the later rate corollary. They are clipped to the canonical ranges. Define \[ F(h,a,b)=\frac{\delta ha+(1-h)b}{d(h)}, \] \[ S_{h,a,b}(O)=F(h,a,b) +\frac{\delta W}{d(h)}(Y-a)+\frac{1-W}{d(h)}(Y-b) +\frac{\delta(a-b)}{d(h)^2}(W-h). \tag{1} \] All functions are evaluated at $X$. The estimator is the average of (1) on an independent evaluation sample, projected onto $[0,1]$.Proposition: Put $r=h-e$, $\Delta_1=a-m_1$, $\Delta_0=b-m_0$. Conditionally on the pilots, the exact bias is \[
B=\mathbb E\left[
\frac{\delta(\delta-1)r^2(m_1-m_0)}{d(h)^2d(e)}
+\frac{(\delta-1)r\{\delta h\Delta_1+(1-h)\Delta_0\}}
{d(h)^2}\right]. \tag{2}
\] For $N$ evaluation observations, \[
\mathbb E(\widehat\psi_\delta-\psi_\delta)^2
\le \frac{C_\delta}{N}+C_\delta\mathbb E\left[
\|r\|_2^4+\|r\|_2^2(\|\Delta_1\|_2+\|\Delta_0\|_2)^2\right], \tag{3}
\] where norms use $P_X$ and $C_\delta$ is finite for each fixed $\delta>0$.
Proof: Take expectation in (1) conditional on $X$ and use $\mathbb E[W(Y-a)\mid X]=e(m_1-a)$ and its untreated analogue. The part involving the true outcome regressions is $F(h,m_1,m_0)-\delta r(m_1-m_0)/d(h)^2$. Since \[
q(h)-q(e)=\frac{\delta r}{d(h)d(e)},\quad
q(h)-q(e)-\frac{\delta r}{d(h)^2}
=\frac{\delta(\delta-1)r^2}{d(h)^2d(e)},
\] it gives the first term in (2). The outcome-error coefficient is \[
\frac{r(\delta\Delta_1-\Delta_0)}{d(h)}
-\frac{\delta r(\Delta_1-\Delta_0)}{d(h)^2}
=\frac{(\delta-1)r\{\delta h\Delta_1+(1-h)\Delta_0\}}{d(h)^2}.
\] This proves the identity. Because $d(t)\ge\min(1,\delta)$ on $[0,1]$, all summands in (1) are uniformly bounded. Apply Cauchy--Schwarz to (2), then conditional variance decomposition and projection to obtain (3). ∎
Corollary: Suppose the propensity pilot is independent of both outcome pilots and \[
\mathbb E\|h-e\|_2^4\le C n^{-4a_e},\quad
\mathbb E\|h-e\|_2^2\le C n^{-2a_e},\quad
\mathbb E\|a-m_1\|_2^2+\mathbb E\|b-m_0\|_2^2\le C n^{-2a_m}.
\] With $N\asymp n$, risk is at most $C\{n^{-1}+n^{-4a_e}+n^{-2(a_e+a_m)}\}$. Thus $a_e\ge1/4$ and $a_e+a_m\ge1/2$ suffice for $O(n^{-1})$ risk.
Proof: Independence factors the mixed squared-norm expectation in (3), and $(x+y)^2\le2x^2+2y^2$ proves the asserted inequality. ∎
These conditions are achievable, for example, with independent histogram pilots and $a_j=\bar s_j/(2\bar s_j+d)$, $\bar s_j=\min(s_j,1)$. Here are details sufficient to justify the moment assertion. In a cube of side $h$, conditional regression oscillation is $O(h^{\bar s})$ and the selected sample count is binomial with success probability at least $c\varepsilon h^d$. A bounded sample mean has fourth central moment $O(N^{-2})$ conditional on its count $N>0$. Splitting according to $N\ge n c\varepsilon h^d/2$ and applying the binomial exponential lower-tail bound gives a fourth cell error bounded by $C\{h^{4\bar s}+(nh^d)^{-2}\}$, including empty cells. Jensen's inequality for integration against $P_X$ gives the same bound for $\mathbb E\|\widehat f-f\|_2^4$. The second-moment bound is $C\{h^{2\bar s}+(nh^d)^{-1}\}$. Balancing the two terms gives the stated exponents. This proof uses density bounds, not density smoothness.A slower-rate lower bound
The next theorem assumes the declared radii admit uniform $p$, a propensity in a neighborhood of $1/2$, and two distinct constant outcome means $0<b_0<a_0<1$. These mild interior conditions are explicit because mere nonemptiness of a Hölder ball need not contain this submodel.Theorem: For every $s=s_e>0$ under the stated interior conditions, there is $c_*>0$ independent of $n$ such that \[
R_{n,\delta}^{\star}\ge c_*\left\{n^{-1}
\mathbin{\vee} n^{-8s/(4s+d)}\right\}. \tag{4}
\] The second term holds even when $p=1$, $m_1=a_0$, $m_0=b_0$ are supplied to the estimator. It is slower than $n^{-1}$ when $s<d/4$.
Proof: Choose a nonzero smooth function $K$ supported strictly inside the unit cube, with integral zero. Tile the cube into $M=h^{-d}$ cubes and let $z_j$ be their lower corners. For independent signs $\omega_j\in\{-1,1\}$ define \[
f_\omega(x)=b h^s\sum_{j=1}^M\omega_jK((x-z_j)/h),\qquad
e_\omega(x)=\tfrac12+f_\omega(x).
\] For fixed sufficiently small $b>0$ these functions have the required Hölder radius and overlap, uniformly in $h$: derivatives scale as $h^{s-k}$, the last derivative has its required Hölder bound, and the compact interior support lets the pieces join with zero derivatives. Use independent Bernoulli outcomes with means $a_0,b_0$ in the two arms. The likelihood relative to $e_*=1/2$ depends only on $(X,W)$. For one observation its cross moment is \[
\mathbb E_*L_\omega L_{\omega'}
=1+4\int f_\omega f_{\omega'}
=1+t_0\sum_j\omega_j\omega_j',\quad
t_0=4b^2h^{2s+d}\int K^2.
\] Let $\overline P^n$ be the uniform mixture of the $2^M$ sample laws. Using $1+x\le e^x$ and independence of the signs, \[
1+\chi^2(\overline P^n,P_*^n)
\le\{\cosh(nt_0)\}^M
\le\exp(Mn^2t_0^2/2)
\le\exp(Cb^4n^2h^{4s+d}). \tag{5}
\] Take integer reciprocal $h\asymp n^{-2/(4s+d)}$ and choose $b$ so that (5) implies total variation at most $1/4$. Now $q''(t)=-2\delta(\delta-1)/d(t)^3$ has fixed nonzero sign and absolute value bounded above and below on the overlap interval. Since $\int f_\omega=0$, Taylor's formula with integral remainder shows, for every sign vector, that the target difference has one common sign and \[
c h^{2s}\le |\psi_\delta(P_\omega)-\psi_\delta(P_*)|
\le C h^{2s}. \tag{6}
\] Indeed $\int f_\omega^2=b^2h^{2s}\int K^2$ and the target multiplier is $a_0-b_0>0$. A test that places an estimated target on the null or the alternative side of the midpoint in (6) has squared error at least $c^2h^{4s}/4$ whenever it errs. The sum of its null error and mixture alternative error is at least $1-\operatorname{TV}\ge3/4$. Hence the maximum risk is at least a constant times $h^{4s}=n^{-8s/(4s+d)}$. Finally vary a constant outcome mean by $\pm b'/\sqrt n$ while keeping $e=1/2,p=1$. The target changes by a nonzero constant times $n^{-1/2}$ and sample KL is bounded by $C(b')^2$. The identical two-point testing argument gives the first term of (4). ∎
Comparison and unresolved region
For a supplied fixed policy $q_0(x)$, the usual one-step bias is a sum of propensity-error times outcome-error products. When both outcome means are known constants and $p$ is uniform, that fixed-policy target can be computed without data. In contrast (4) persists for the incremental policy because the unknown propensity enters the estimand itself. Equation (2) identifies the corresponding squared-propensity remainder. At $\delta=1$ both remainders vanish and (1) reduces exactly to $Y$, consistent with the excluded sample-mean case. The lower bound (4) and the first-order upper bound do not generally match. The remaining task is to construct higher-order estimators and mixed propensity--outcome lower bounds, including the role of unknown $p$ and all smoothness boundaries. No full minimax formula is claimed here.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.
- [kennedy] E. H. Kennedy. Nonparametric causal effects based on incremental propensity score interventions. Journal of the American Statistical Association 114 (2019), 645--656. https://arxiv.org/abs/1704.00211.