Sharp Ordinal Benefit Bounds

The potential treatments $D_0,D_1$, potential selection indicators $S_0,S_1$, and potential ordered outcomes $Y_0,Y_1$ are conditionally independent of the binary instrument $Z$ given the covariate cell $X$: $$(D_0,D_1,S_0,S_1,Y_0,Y_1)\perp Z\mid X.$$

For every covariate cell $x$ with $P(C\mid X=x)>0$, the covariate-specific direction $d(x)\in\{-1,+1\}$ satisfies $$P\{d(x)(S_1-S_0)\ge 0\mid C,X=x\}=1.$$

Let $P_{\mathrm{obs}}$ be the observed-data law generated by a five-node potential-outcome slate with ordered outcome support of size $K\ge 3$. Suppose the latent law satisfies the structural restrictions in Definition P-1 for the instrument-overlap constant $\varepsilon_Z$ and weak-selection direction $d(x)$. Let $c$ be the observable capacity vector from Definition P-3, identified from $P_{\mathrm{obs}}$ as in Theorem T-1, and suppose that $$M=\sum_{x\in\mathcal X}p_xm(x)>0.$$ Then the following statements hold.

• (Sharp cellwise projection.) For every cell $x$ with $p_x>0$, the set of survivor-complier couplings generated by feasible full laws with observed law $P_{\mathrm{obs}}$ is exactly $$\Gamma_x^{\ast} = \left\{\gamma_x\ge0: \sum_j\gamma_{ij,x}\le \ell_i(x),\ \sum_i\gamma_{ij,x}\le h_j(x),\ \sum_{i,j}\gamma_{ij,x}=m(x) \right\}.$$
• (Exact-mass branch.) For every cell $x$ with $p_x>0$, $$\Delta q(x)>0 \Rightarrow \Gamma_x^{\ast}=\Gamma_x^{+},\qquad \Delta q(x)<0 \Rightarrow \Gamma_x^{\ast}=\Gamma_x^{-},\qquad \Delta q(x)=0 \Rightarrow \Gamma_x^{\ast}=\Gamma_x^{0}.$$
• (Threshold endpoints.) For every cell $x$ with $p_x>0$, $$\inf_{\gamma_x\in\Gamma_x^{\ast}} b_x(\gamma_x)=B_L(x), \qquad \sup_{\gamma_x\in\Gamma_x^{\ast}} b_x(\gamma_x)=B_U(x).$$ Writing $\gamma_x^L$ and $\gamma_x^U$ for the lower and upper cellwise threshold-flow couplings from Definition P-7, $$\gamma_x^L\in\Gamma_x^{\ast},\qquad b_x(\gamma_x^L)=B_L(x),\qquad \gamma_x^U\in\Gamma_x^{\ast},\qquad b_x(\gamma_x^U)=B_U(x).$$
• (Zero-mass cells.) For every cell $x$ with $m(x)=0$, $$\Gamma_x^{\ast}=\{0\}.$$

Consequently, the sharp identified interval for the benefit probability over feasible full laws is $$\Theta_I(P_{\mathrm{obs}}) = \left[ \frac{\sum_{x\in\mathcal X}p_xB_L(x)}{M}, \frac{\sum_{x\in\mathcal X}p_xB_U(x)}{M} \right].$$

For every $x\in\mathcal X$, define the prefix and tail sums $$\ell_{\le t}(x)=\sum_{i\le t}\ell_i(x), \qquad h_{\le t}(x)=\sum_{j\le t}h_j(x),$$ $$\ell_{<t}(x)=\sum_{i<t}\ell_i(x), \qquad h_{>t}(x)=\sum_{j>t}h_j(x).$$ The lower and upper threshold-cut values are $$B_L(x) = \max\left\{ 0,\ \max_{t\in\mathcal Y}\bigl[\ell_{\le t}(x)-h_{\le t}(x)\bigr] +\min\{\Delta q(x),0\} \right\}$$ and $$B_U(x) = \min\left\{ m(x),\ \min_{t\in\mathcal Y}\bigl[\ell_{<t}(x)+h_{>t}(x)\bigr] \right\}.$$ These formulas apply on positive-survivor zero-gap cells and on zero-survivor cells.

Given observations $O_1,\ldots,O_n$, a finite covariate support $\mathcal X$, outcome support $\mathcal Y=\{0,\ldots,K-1\}$, and screening threshold $\eta_n$, define the projected screened plug-in endpoint estimator $\widehat\Psi_n$ by the following steps.

1. For each empirical conditional probability given $(X=x,Z=z)$, use the empirical ratio when the empirical denominator is positive and use the fixed value zero when that denominator is zero. These empirical conditional probabilities form the raw capacity vector $$\widetilde c_n=(\widetilde\ell,\widetilde h).$$
2. Define the nonnegative capacity cone and the projected empirical capacity vector by $$\mathfrak C=\prod_{x\in\mathcal X}\mathbb R_+^{2K}, \qquad \widehat c_n =\Pi_{\mathfrak C}(\widetilde c_n) =\arg\min_{c\in\mathfrak C}\|c-\widetilde c_n\|_2^2.$$ With the fixed Euclidean metric, the minimizer is unique, and $\Pi_{\mathfrak C}$ is the coordinatewise positive-part map. Write $$\widehat c_n=(\widehat\ell,\widehat h).$$
3. For each $x\in\mathcal X$, define the empirical capacity totals, capacity-total gap, and empirical exact mass by $$\widehat q_0(x)=\sum_i\widehat\ell_i(x), \qquad \widehat q_1(x)=\sum_j\widehat h_j(x),$$ $$\widehat{\Delta q}(x)=\widehat q_1(x)-\widehat q_0(x), \qquad \widehat m(x)=\min\{\widehat q_0(x),\widehat q_1(x)\}.$$
4. For each $x\in\mathcal X$, define the empirical lower and upper threshold-cut values by $$\widehat B_L(x)= \max\left\{ 0,\ \max_{t\in\mathcal Y} \left[ \sum_{i\le t}\widehat\ell_i(x) - \sum_{j\le t}\widehat h_j(x) \right] + \min\{\widehat{\Delta q}(x),0\} \right\}$$ and $$\widehat B_U(x)= \min\left\{ \widehat m(x),\ \min_{t\in\mathcal Y} \left[ \sum_{i<t}\widehat\ell_i(x) + \sum_{j>t}\widehat h_j(x) \right] \right\}.$$
5. Define the empirical retained-cell mass score and retained-cell indicator by $$\widehat r_x=\widehat p_x\widehat m(x), \qquad \widehat I_x=\mathbf 1\{\widehat r_x>\eta_n\}.$$
6. Define the screened empirical aggregate mass and screened empirical endpoint numerators by $$\widehat M_n=\sum_x \widehat I_x\,\widehat p_x\widehat m(x),$$ $$\widehat N_{L,n}=\sum_x \widehat I_x\,\widehat p_x\widehat B_L(x), \qquad \widehat N_{U,n}=\sum_x \widehat I_x\,\widehat p_x\widehat B_U(x).$$
7. Output the projected screened plug-in endpoint estimator $$\widehat\Psi_n= \begin{cases} \left(\widehat N_{L,n}/\widehat M_n,\ \widehat N_{U,n}/\widehat M_n\right), & \widehat M_n>0,\\ (0,1), & \widehat M_n=0. \end{cases}$$

Fix a slate potential-outcome system with finite covariate support and ordered outcome support of size $K\ge 3$. Let $P_{\mathrm{obs}}$ be its observed-data law, let $O_1,\ldots,O_n$ denote the observed cells, let $d(x)\in\{-1,+1\}$ be the weak selection-monotonicity direction, let $\varepsilon_Z$ be the instrument-overlap constant, and let $\eta_n$ be the screening threshold from Definition P-9. Suppose that:

• (Sampling.) For every $n$, the first $n$ observations satisfy Assumption A-11 with common law $P_{\mathrm{obs}}$.
• (Overlap.) The instrument propensity satisfies Assumption A-5 with constant $\varepsilon_Z$.
• (Structural restrictions.) The latent law satisfies Assumption A-1, Assumption A-2, Assumption A-3, Assumption A-4, Assumption A-6, Assumption A-7, Assumption A-9, with weak selection monotonicity in direction $d$.
• (Screening rate.) For every $n$, $\eta_n>0$, with $\eta_n\to0$ and $\sqrt n\,\eta_n\to\infty$.
• (Limit inputs.) The finite multinomial central limit theorem applies to the empirical observed-cell vector, and the Hadamard directional delta method applies to tangentially directionally differentiable maps.

Let $c$ be the observable capacity vector from Definition P-3. Let $$M=\sum_x p_x m(x),$$ where $m(x)$ is the exact survivor-complier mass in cell $x$, and let $\mathcal X_+(P)=\{x:p_xm(x)>0\}$. Let $\Psi(P_{\mathrm{obs}})=(\theta_L,\theta_U)$ be the endpoint map identified by Theorem T-1, and let $\widehat\Psi_n$ be the projected screened plug-in estimator from Definition P-9. Then $$P\!\left(\{x:\widehat r_x>\eta_n\}=\mathcal X_+(P)\right)\to 1$$ and $$\widehat\Psi_n\to_P\Psi(P_{\mathrm{obs}}).$$ Moreover, there exist a Gaussian law $\mathbb Z_{P_{\mathrm{obs}}}$ on observed-cell score functions and a map $D\Psi$ such that $$\int z\,d\mathbb Z_{P_{\mathrm{obs}}}(z)=0$$ and, for observed cells $a,b$, $$\int z(a)z(b)\,d\mathbb Z_{P_{\mathrm{obs}}}(z) = \begin{cases} P_{\mathrm{obs}}(a)\{1-P_{\mathrm{obs}}(a)\}, & a=b,\\ -P_{\mathrm{obs}}(a)P_{\mathrm{obs}}(b), & a\ne b. \end{cases}$$ The map $D\Psi$ is the Hadamard directional derivative, at the observed-law mass vector and on the recovered support $\mathcal X_+(P)$, of the reduced-support endpoint map. With this derivative, $$\sqrt n\{\widehat\Psi_n-\Psi(P_{\mathrm{obs}})\} \rightsquigarrow D\Psi(\mathbb Z_{P_{\mathrm{obs}}}).$$