Honest Length for Transported Complier Effects

Fix $n$, $P$, $N_n$, $k_n$, $c$, and the instrument-overlap constant $\varepsilon$. Suppose that:

• (Model class.) $P$ belongs to the transported complier-effect model class $\mathcal P_n$ of Definition P-1.
• (Source contrasts.) The source conditional outcome contrast $\Delta_Y(x)$ and the source conditional receipt contrast $\Delta_D(x)$ are measurable and integrable with respect to the source covariate marginal $P_S^X$.

Then $\Delta_Y$ and $\Delta_D$ represent the observed source contrasts: for every measurable covariate set $A$, $$\int_{\{O^S:X\in A\}} H(O^S)Y\,dP_S = \int_A \Delta_Y(x)\,dP_S^X(x), \qquad \int_{\{O^S:X\in A\}} H(O^S)D\,dP_S = \int_A \Delta_D(x)\,dP_S^X(x).$$ Moreover, with $$\mu_{Y,n} = \int \bigl(Y(1)-Y(0)\bigr)\mathbf 1\{D(1)=1,\ D(0)=0\}\,dP_n^F(\,\cdot\mid S=0)$$ and $$\mu_n = P_n^F\{D(1)=1,\ D(0)=0\mid S=0\},$$ the transported reduced-form and first-stage means satisfy $$\int w(x)\Delta_Y(x)\,dP_S^X(x)=\mu_{Y,n}, \qquad \int w(x)\Delta_D(x)\,dP_S^X(x)=\mu_n.$$ Consequently, $$\frac{\mu_{Y,n}}{\mu_n} = \theta_T = \mathbb E_{P_n^F}\!\left[Y(1)-Y(0)\mid D(1)=1,\ D(0)=0,\ S=0\right] \in \Theta=[-1,1].$$

Fix $c>0$, an instrument-overlap constant $\varepsilon\in(0,1/2)$, and a noncoverage level $\alpha\in(0,1)$. There exist constants $c_0>0$ and $t_c>0$, depending only on $(\alpha,\varepsilon,c)$, such that the following holds for every measurable covariate space $\mathcal X$. Let $N_n$ and $k_n$ be integer sequences satisfying:

• (Target-source ratio.) $N_n/n\to c$.
• (Positive cells.) $k_n>0$ for every $n$.
• (Diverging cells.) $k_n\to\infty$.
• (Sub-root growth.) $k_n/\sqrt n\to 0$.
• (Measurable carrier.) For each $n$, $\mathcal X$ contains a measurable $k_n$-cell carrier.
• (Transported-IV model.) Every array in the transported complier-effect class $\mathcal P_n$ of Definition P-1 satisfies Assumption A-1, Assumption A-2, Assumption A-3, Assumption A-4, Assumption A-5, Assumption A-6, Assumption A-7, Assumption A-8, Assumption A-9, Assumption A-10, Assumption A-11, Assumption A-12, Assumption A-13, Assumption A-14, Assumption A-15.
• (Finite-cell source.) Every array in the finite-cell class $\mathcal N_n$ of Definition P-2 satisfies Assumption A-16.

Then, for every effective-strength threshold $t_0>0$, $$c_0 \min\{1,t_0^{-1/2}\} \le V^\star(t_0)$$ and $$c_0 \min\{1,t_0^{-1/2}\} \le V_{\mathcal N}^\star(t_0),$$ where $V^\star(t_0)$ and $V_{\mathcal N}^\star(t_0)$ are the oracle values in Definition P-5, Definition P-6. Moreover, whenever $0<t_0\le t_c$, $$c_0 \le V^\star(t_0).$$

Let $N_n$ and $k_n$ be integer sequences, and let $c$, $\varepsilon$, and $\alpha$ be constants. Suppose that:

• (Sampling ratio.) $c>0$ and $N_n/n\to c$.
• (Overlap and coverage constants.) $0<\varepsilon<1/2$ and $0<\alpha<1$.
• (Envelope scale.) $k_n>0$ for every $n$, $k_n\to\infty$, and $k_n/\sqrt n\to0$.
• (Admissible geometry.) The measurable covariate space admits a deterministic geometry $\mathfrak g$ satisfying Definition P-11 with envelope scale $k_n$ and overlap constant $\varepsilon$.
• (Model-class implications.) For every $n$ and every transported array $P$, membership of $P$ in the transported complier-effect model class $\mathcal P_n$ of Definition P-1 implies Assumption A-3, Assumption A-4, Assumption A-13, Assumption A-14, Assumption A-15.

Define $$L_\alpha=\left(\frac{8}{\alpha\varepsilon^2}\right)^{1/2}, \qquad H_i=\frac{Z_i}{e(X_i)}-\frac{1-Z_i}{1-e(X_i)},$$ and $$\widehat A_n=\frac1n\sum_{i=1}^n w(X_i)H_iY_i, \qquad \widehat B_n=\frac1n\sum_{i=1}^n w(X_i)H_iD_i, \qquad \widehat\kappa_n=\frac1n\sum_{i=1}^n w(X_i)^2 .$$ Let the score-inversion confidence set be $$C_n = \left\{\vartheta\in\Theta: \left|\widehat A_n-\vartheta\widehat B_n\right| \le L_\alpha\sqrt{\widehat\kappa_n/n} \right\}.$$ Then the sequence $(C_n)$ is oracle honest, $(C_n)\in\mathfrak C^{\mathrm{or}}$ in the sense of Definition P-3, and for every fixed effective-strength threshold $t_0>0$, $$R((C_n),t_0) \le C_0\min\{1,t_0^{-1/2}\}, \qquad C_0=\max\left\{2,4L_\alpha+\frac8{\varepsilon^2}\right\}.$$ Consequently, for the oracle minimax value $V^\star(t_0)$ of Definition P-5, $$\frac{3(1-\alpha)^2}{16}\min\{1,t_0^{-1/2}\} \le V^\star(t_0) \le C_0\min\{1,t_0^{-1/2}\},$$ and for every admissible geometry $\mathfrak g'\in\mathcal G$ the fixed-geometry value of Definition P-14 obeys the same two-sided bounds, $$\frac{3(1-\alpha)^2}{16}\min\{1,t_0^{-1/2}\} \le V_{\mathfrak g'}^\star(t_0) \le C_0\min\{1,t_0^{-1/2}\}.$$

Let $c>0$, let $N_n$ be target sample sizes with $N_n/n\to c$, let $k_n$ be positive integers with $k_n\to\infty$ and $k_n/\sqrt n\to0$, let $\varepsilon\in(0,1/2)$, and let $\alpha\in(0,1)$. Let $\mathfrak g$ be a deterministic geometry array satisfying Definition P-11 at scale $k_n$ and overlap parameter $\varepsilon$. Assume that, for every $n$, every law in the fixed-geometry slice $\mathcal P_n(\mathfrak g)$ of Definition P-12 satisfies

• (Two samples.) Assumption A-3 with target-to-source ratio $c$.
• (Overlap.) Assumption A-4 with overlap parameter $\varepsilon$.
• (Envelope.) Assumption A-13 at scale $k_n$.
• (Second moment.) Assumption A-14 at scale $k_n$.
• (Degradation.) Assumption A-15 at scale $k_n$.

Define $$L_\alpha=\left(\frac{8}{\alpha\varepsilon^2}\right)^{1/2}, \qquad c_\alpha=\frac{3(1-\alpha)^2}{16}, \qquad C_\alpha=\max\{2,4L_\alpha+8\varepsilon^{-2}\}.$$ Then, for every fixed effective-strength threshold $t_0>0$, the fixed-geometry minimax expected-length value $V_{\mathfrak g}^\star(t_0)$ of Definition P-14 satisfies $$c_\alpha\min\{1,t_0^{-1/2}\} \le V_{\mathfrak g}^\star(t_0) \le C_\alpha\min\{1,t_0^{-1/2}\}.$$ The constants $c_\alpha$ and $C_\alpha$ depend only on $\alpha$ and $\varepsilon$, uniformly over admissible deterministic geometry arrays $\mathfrak g$.

Let $N_n$ and $k_n$ be positive integer sequences, let $c,\varepsilon,\alpha\in\mathbb R$, and suppose the following conditions hold.

• (Sampling ratio.) $c>0$ and $N_n/n\to c$.
• (Overlap and coverage levels.) $0<\varepsilon<1/2$ and $0<\alpha<1$.
• (Cell growth.) $k_n>0$ for every $n$, $k_n\to\infty$, and $k_n/\sqrt n\to0$.
• (Ambient carrier.) For every $n$, the measurable covariate space contains an injected copy of $\{1,\ldots,k_n\}$ whose singleton images are measurable.
• (Finite-cell rows.) For every row $n$ and every transported array $P\in\mathcal N_n$, with $\mathcal N_n$ as in Definition P-2, the row satisfies the two-sample condition in Assumption A-3, the uniform finite-cell source condition in Assumption A-16, and the degrading-array condition in Assumption A-15.

Put $$B_c=32(1+c^{-1}),\qquad L_{\alpha,c}=\left(\frac{2B_c}{\alpha}\right)^{1/2},\qquad C_0=\max\{2,4\sqrt2\,L_{\alpha,c}+8B_c\}.$$ Then $C_0>0$, and there exists a sample-only finite-cell confidence-set sequence $C=(C_n)$ in the feasible class $\mathfrak C^{\mathrm{cell}}$ of Definition P-7 such that, for every $n$ and every $P\in\mathcal N_n$, $C_n$ agrees $P$-almost surely under the two-sample law with the finite-cell score-inversion rule using critical constant $L_{\alpha,c}$. Equivalently, on the finite-cell experiment, this rule is given by $$H_i=2(2Z_i-1),\qquad \widehat p_{x,n}=\frac1{N_n}\sum_{j=1}^{N_n}\mathbf 1\{X_j^T=x\},$$ and, for $G\in\{Y,D\}$, $$\widehat m_{G,n}(x)=\frac{k_n}{n}\sum_{i=1}^n \mathbf 1\{X_i=x\}H_iG_i,\qquad \widehat M_{G,n}=\sum_{x=1}^{k_n}\widehat p_{x,n}\widehat m_{G,n}(x).$$ For $N_n\ge2$, define $$\widehat\kappa_{U,n} =\frac{k_n}{N_n(N_n-1)} \sum_{j\ne\ell}\mathbf 1\{X_j^T=X_\ell^T\},\qquad \widehat K_n=1+\widehat\kappa_{U,n},$$ and set $$C_n=\left\{\vartheta\in\Theta: |\widehat M_{Y,n}-\vartheta\widehat M_{D,n}| \le L_{\alpha,c}\sqrt{\widehat K_n/n}\right\},$$ with $C_n=\Theta$ at indices for which $N_n<2$. For every $t_0>0$, $$\limsup_{n\to\infty}\sup_{P\in\mathcal N_n:\,t_n\ge t_0} \mathbb E_P[\lambda(C_n)] \le C_0\min\{1,t_0^{-1/2}\}.$$ Moreover, for every $t_0>0$, the feasible finite-cell honest minimax risk satisfies $$\frac{3(1-\alpha)^2}{16}\min\{1,t_0^{-1/2}\} \le \inf_{\widetilde C\in\mathfrak C^{\mathrm{cell}}} \limsup_{n\to\infty}\sup_{P\in\mathcal N_n:\,t_n\ge t_0} \mathbb E_P[\lambda(\widetilde C_n)].$$

Fix $c,\varepsilon,\alpha,c_-,c_+\in\mathbb R$, target sample sizes $N_n$, and cell counts $k_n$. Suppose that

• (Sample-size ratio.) $0<c$ and $N_n/n\to c$.
• (Overlap and coverage levels.) $0<\varepsilon<1/2$ and $0<\alpha<1$.
• (Regular cell constants.) $0<c_-\le 1$ and $1\le c_+$.
• (Cell growth.) $k_n>0$ for every $n$, $k_n\to\infty$, and $k_n/\sqrt n\to0$.
• (Measurable carrier.) For every $n$, the covariate carrier contains $k_n$ injected measurable singleton cells.
• (Model restrictions.) Every array $P\in\mathcal N_n^{\mathrm{reg}}$ from Definition P-15 with constants $c,\varepsilon,c_-,c_+$ satisfies Assumption A-3, Assumption A-4, and Assumption A-15.

On $\mathcal N_n^{\mathrm{reg}}$, let $q_{x,n}=P_S^X(X=x)$ and define $$H_i=\frac{Z_i}{e(X_i)}-\frac{1-Z_i}{1-e(X_i)}.$$ For $G\in\{Y,D\}$, set $$\widehat p_{x,n}=\frac1{N_n}\sum_{j=1}^{N_n}\mathbf 1\{X_j^T=x\}, \qquad \widehat m_{G,n}(x)=\frac1{nq_{x,n}}\sum_{i=1}^n \mathbf 1\{X_i=x\}H_iG_i,$$ and $$\widehat M_{G,n}=\sum_{x=1}^{k_n}\widehat p_{x,n}\widehat m_{G,n}(x).$$ For $N_n\ge2$, set $$\widehat\kappa_{U,n} = \frac1{N_n(N_n-1)}\sum_{j\ne\ell} \frac{\mathbf 1\{X_j^T=X_\ell^T\}}{q_{X_j^T,n}}, \qquad \widehat K_n=1+\widehat\kappa_{U,n},$$ and put $C_n^{\mathrm{reg}}=\Theta$ when $N_n<2$. Let $$B_{\varepsilon,c}=8(\varepsilon^{-2}+c^{-1}).$$ For every $L\ge\{2B_{\varepsilon,c}/\alpha\}^{1/2}$, define $$C_{0}=\max\{2,4\sqrt2L+8B_{\varepsilon,c}\}.$$ Then $0<C_0$, and there exists a regular-cell procedure $C=(C_n)_{n\ge1}$ such that, for every $n$ and every $P\in\mathcal N_n^{\mathrm{reg}}$, $C_n$ agrees almost surely under the two-sample law with $$C_n^{\mathrm{reg}} = \left\{\vartheta\in\Theta: |\widehat M_{Y,n}-\vartheta\widehat M_{D,n}| \le L\sqrt{\widehat K_n/n}\right\}$$ for $N_n\ge2$, together with the preceding convention for $N_n<2$. This procedure belongs to $\mathfrak C^{\mathrm{reg}}$ from Definition P-16. Moreover, for every fixed $t_0>0$, $$\limsup_{n\to\infty}\sup_{P\in\mathcal N_n^{\mathrm{reg}}:\,t_n\ge t_0} \mathbb E_P[\lambda(C_n)] \le C_0\min\{1,t_0^{-1/2}\}.$$ For every fixed $t_0>0$, the minimax risk over $\mathfrak C^{\mathrm{reg}}$ satisfies $$\frac{3(1-\alpha)^2}{16}\min\{1,t_0^{-1/2}\} \le \inf_{C\in\mathfrak C^{\mathrm{reg}}} \limsup_{n\to\infty}\sup_{P\in\mathcal N_n^{\mathrm{reg}}:\,t_n\ge t_0} \mathbb E_P[\lambda(C_n)].$$ Thus, with known regular source-cell probabilities and a known cell-varying propensity satisfying the stated overlap, the feasible regular-cell construction attains the oracle frontier order on $\mathcal N_n^{\mathrm{reg}}$.