Minimax Inference for Threshold Clamp Policies

For $P\in\mathcal M$ as in Definition P-1, $x\in\mathcal X$, and $0\leq\delta\leq\bar\delta$, define $$q_x(\delta)=\int_{[0,\delta]}\pi_x(a)\,\mathrm da, \qquad \nu_\delta(P)=\mathbb E_P\!\left[Y\mathbf 1\{A>\delta\}\right].$$ The unsmoothed observed-data clamp target is $$\theta_\delta(P) = \nu_\delta(P) + \sum_{x\in\mathcal X}p_xq_x(\delta)\mu_x^P(\delta).$$

For every $x\in\mathcal X$ and for Lebesgue-almost every $a\in[0,1]$, $$c_- a^\kappa \leq \pi_x(a) \leq c_+ a^\kappa .$$

The regression function $\mu_x^P$ satisfies the following condition with parameters $\beta$ and $L$. For every $x\in\mathcal X$:

• (Continuity.) The map $a\mapsto \mu_x^P(a)$ is continuous on $[0,1]$.
• (Range.) For every $a\in[0,1]$, $\mu_x^P(a)\in[0,1]$.
• (Regression version.) The function $\mu_X^P(A)$ realizes the conditional outcome regression given the complete design: $$\mathbb E_P[Y\mid X,A]=\mu_X^P(A) \quad P\text{-almost surely}.$$
• (Hölder Taylor remainder.) Let $\ell=\ell(\beta)$ be the local-polynomial order defined in Definition. For all $s,t\in[0,1]$, with derivatives taken intrinsically on $[0,1]$, $$\left| \mu_x^P(t) - \sum_{j=0}^{\ell} \frac{(\mu_x^P)^{(j)}(s)}{j!}(t-s)^j \right| \le L|t-s|^\beta .$$

The information-balance bandwidth $h_n$ for the threshold sequence $\delta_n$ is $$h_n = \inf\left\{ h\in(0,1-\bar\delta]: n h^{2\beta+1}(\delta_n+h)^\kappa\geq 1 \right\},$$ with $h_n=1-\bar\delta$ when the displayed set is empty.

The Hölder minimax rate $r_n$ is $$r_n=n^{-1/2}+\delta_n^{\kappa+1}h_n^\beta .$$

Fix $J,\beta,\kappa,L,c_-,c_+,p_{\min},\bar\delta,\alpha$ satisfying the regime conditions:

• (Regime.) $J\geq 1$, $\beta>0$, $\kappa\geq0$, $L>0$, $c_->0$, $c_-\leq \kappa+1\leq c_+$, $p_{\min}>0$, $p_{\min}\leq 1/J$, $\bar\delta\in(0,1)$, and $\alpha\in(0,1/2)$.

Then there exist constants $0<c<C$ and an amplitude $a\in(0,1/4]$, depending only on the displayed regime constants, such that, for every deterministic threshold sequence $(\delta_n)_{n\geq1}$ with $\delta_n\in[0,\bar\delta]$ for every $n$ and every deterministic sequence $B_n=(I_{0,n},I_{1,n},I_{2,n})$ of three pairwise-disjoint sample blocks with $|I_{j,n}|\geq \lfloor n/4\rfloor$ for $j=0,1,2$, for all sufficiently large $n$, with $h_n$ the information-balance bandwidth from Definition P-3 and $$r_n=n^{-1/2}+\delta_n^{\kappa+1}h_n^\beta$$ as in Definition P-4, the observed minimax absolute-error risk $R_n^\star$ over the clamp model $\mathcal M$ in Definition P-1 and the worst-case i.i.d. risk over $\mathcal M$ of the stabilized estimator using $B_n$ satisfy $$c r_n \leq R_n^\star \leq \sup_{P\in\mathcal M} \mathbb E_P\!\left[ \left|\widehat\theta_{n,B_n}^{\mathrm{TG}}-\theta_{\delta_n}(P)\right| \right] \leq C r_n .$$ Moreover, for all sufficiently large $n$, the lower bound is witnessed inside the same clamp model by:

• (Global shift.) Laws $P_0,P_1\in\mathcal M$ satisfying Assumption A-1, with the same $(X,A)$-design distribution and the same $p_x$ and $\pi_x$ functions, having Bernoulli outcomes, and obeying $$\mu_x^{P_1}(a)=\mu_x^{P_0}(a)+n^{-1/2} \quad\text{for every }x\text{ and }a\in[0,1],$$ with well-posed product chi-squared divergence, $$P_1^{\otimes n}\ll P_0^{\otimes n}, \qquad \int \left(\frac{dP_1^{\otimes n}}{dP_0^{\otimes n}}-1\right)^2\,dP_0^{\otimes n}<\infty,$$ and $$c n^{-1/2} \leq \left|\theta_{\delta_n}(P_1)-\theta_{\delta_n}(P_0)\right|, \qquad \chi^2\!\left(P_1^{\otimes n},P_0^{\otimes n}\right)\leq C .$$
• (Localized perturbation.) Laws $Q_0,Q_1\in\mathcal M$ satisfying Assumption A-1, with the same $(X,A)$-design distribution and the same $p_x$ and $\pi_x$ functions, having Bernoulli outcomes, and admitting a continuous bump $b:\mathbb R\to\mathbb R$ satisfying $b(0)=1$, $b(u)\geq0$ for every $u$, $b(u)=0$ for $u\notin[-1,1]$, and $|b(u)|\leq1$ for $u\in[-1,1]$, with regression shift $$\mu_x^{Q_1}(t)-\mu_x^{Q_0}(t) = a\,h_n^\beta b\!\left(\frac{t-\delta_n}{h_n}\right)$$ for every $x$ and $t\in[0,1]$. The product chi-squared divergence is well posed, $$Q_1^{\otimes n}\ll Q_0^{\otimes n}, \qquad \int \left(\frac{dQ_1^{\otimes n}}{dQ_0^{\otimes n}}-1\right)^2\,dQ_0^{\otimes n}<\infty,$$ and $$c\,\delta_n^{\kappa+1}h_n^\beta \leq \left|\theta_{\delta_n}(Q_1)-\theta_{\delta_n}(Q_0)\right|, \qquad \chi^2\!\left(Q_1^{\otimes n},Q_0^{\otimes n}\right)\leq C .$$

There are constants $c,C\in(0,\infty)$, with $c<C$, for which the following holds.

• (Regime constants.) The number of strata $J$ is a positive integer, $\beta>0$, $\kappa\geq0$, $L>0$, $c_->0$, $c_-\leq \kappa+1\leq c_+$, $p_{\min}>0$, $p_{\min}\leq 1/J$, $\bar\delta\in(0,1)$, and $\alpha\in(0,1/2)$.
• (Threshold path.) The deterministic thresholds satisfy $0\leq \delta_n\leq \bar\delta$ for every $n$.
• (Sample splits.) For every $n$, $B_n=(I_{0,n},I_{1,n},I_{2,n})$ is a deterministic three-way split of the sample indices with pairwise disjoint blocks and $|I_{j,n}|\geq \lfloor n/4\rfloor$ for $j=0,1,2$.

For all sufficiently large $n$, set $h_n$ to be the information-balance bandwidth for $\delta_n$ from Definition P-3, and set $$r_n=n^{-1/2}+\delta_n^{\kappa+1}h_n^\beta .$$ Then the stabilized interval $\mathrm{CI}_n$ has worst-case coverage $$\inf\Bigl\{ P^{\otimes n}\{\theta_{\delta_n}(P)\in \mathrm{CI}_n(O_1,\ldots,O_n)\}: P\in\mathcal M \Bigr\}\geq 1-\alpha,$$ and its worst-case expected length is comparable to $r_n$: $$c\,r_n \leq \sup\Bigl\{ \mathbb E_{P^{\otimes n}}\bigl[\operatorname{len}(\mathrm{CI}_n)\bigr]: P\in\mathcal M \Bigr\} \leq C\,r_n .$$ Moreover, the minimax honest expected length satisfies $$c\,r_n \leq L_n^\star,$$ where $L_n^\star$ is the infimum, over observed-sample confidence procedures with uniform coverage at least $1-\alpha$ over $\mathcal M$, of their worst-case expected length. Equivalently, every observed-sample confidence procedure $C_n$ with $$\inf\Bigl\{ P^{\otimes n}\{\theta_{\delta_n}(P)\in C_n(O_1,\ldots,O_n)\}: P\in\mathcal M \Bigr\}\geq 1-\alpha$$ has worst-case expected length at least $c\,r_n$: $$c\,r_n \leq \sup\Bigl\{ \mathbb E_{P^{\otimes n}}\bigl[\operatorname{len}(C_n)\bigr]: P\in\mathcal M \Bigr\}.$$

Let $J$ be a positive integer and let $$\beta>0,\qquad \kappa\geq 0,\qquad L>0,\qquad 0<c_-\leq \kappa+1\leq c_+,\qquad 0<p_{\min}\leq 1/J,\qquad 0<\bar\delta<1,\qquad 0<\alpha<1/2 .$$ Define the phase and edge scales by $$\delta_{\mathrm{crit},n} = n^{-1/\{2(\beta\kappa+2\beta+\kappa+1)\}}, \qquad \delta_{\mathrm{edge},n} = n^{-1/(2\beta+\kappa+1)} .$$ Then $$\frac{\delta_{\mathrm{crit},n}}{\delta_{\mathrm{edge},n}}\to\infty .$$ Moreover, for every deterministic threshold sequence $(\delta_n)_{n\geq 1}$ with $\delta_n\in[0,\bar\delta]$ for every $n$, let $h_n$ be the information-balance bandwidth in Definition P-3, and set $$a_n=\delta_n^{\kappa+1}h_n^\beta, \qquad r_n=n^{-1/2}+\delta_n^{\kappa+1}h_n^\beta$$ as in Definition P-4. The following conclusions hold:

• Regular thresholds. If $\delta_n/\delta_{\mathrm{crit},n}\to 0$, then $r_n\asymp n^{-1/2}$.
• Critical thresholds. For every $c_0>0$, if $\delta_n/\delta_{\mathrm{crit},n}\to c_0$, then $a_n\asymp n^{-1/2}$ and $r_n\asymp n^{-1/2}$.
• Vanishing atom-dominated thresholds. If $\delta_n/\delta_{\mathrm{crit},n}\to\infty$ and $\delta_n\to0$, then $$r_n\asymp \delta_n^{\kappa+1}\bigl(n\delta_n^\kappa\bigr)^{-\beta/(2\beta+1)} .$$
• Fixed thresholds. For every $\delta_0\in(0,\bar\delta]$, if $\delta_n\to\delta_0$, then $$r_n\asymp n^{-\beta/(2\beta+1)} .$$
• Zero threshold identities. For every $n$ and every $P\in\mathcal M$ satisfying Definition P-1, $$q_x(0)=0\quad\text{for every }x,\qquad \theta_0(P)=\int Y\,dP,\qquad r_n\big|_{\delta_n=0}=n^{-1/2},$$ where the last identity uses the bandwidth $h_n$ at threshold $0$.
• Zero threshold estimator. For every $n$, every $P\in\mathcal M$ satisfying Definition P-1, and every admissible three-way split $B=(I_0,I_1,I_2)$ as in Definition P-5, the total-Gram estimator in Definition P-5, formed at threshold $0$ with bandwidth $h_n$, agrees under the product sampling law generated by $P$ with $$z\mapsto \operatorname{clamp}_{[0,1]}\!\left(\frac{1}{|I_0|}\sum_{i\in I_0}Y_i\right).$$
• Zero threshold stabilized guarantees. For every deterministic sequence $B_\bullet=(B_n)_{n\geq1}$ of admissible three-way split blocks, the stabilized zero-threshold procedure has coverage at least $1-\alpha$ for all sufficiently large $n$. Its stabilized worst-case risk at threshold $0$ and its stabilized worst-case expected length at threshold $0$ with noncoverage level $\alpha$ are both asymptotic to $n^{-1/2}$.

Let $J\in\mathbb N$, and let $\beta,\kappa,L,c_-,c_+,p_{\min},\bar\delta,\alpha\in\mathbb R$. Suppose that

• (Regime constants.) The constants satisfy $$0<J,\qquad \beta>0,\qquad \kappa\geq0,\qquad L>0,\qquad c_->0,\qquad c_-\leq \kappa+1\leq c_+,$$ $$p_{\min}>0,\qquad p_{\min}\leq \frac1J,\qquad 0<\bar\delta<1,\qquad 0<\alpha<\frac12 .$$
• (Threshold sequence.) The deterministic thresholds $(\delta_n)_{n\in\mathbb N}$ satisfy $$0\leq \delta_n\leq \bar\delta$$ for every $n\in\mathbb N$.
• (Split blocks.) For each $n\in\mathbb N$, $B_n=(I_{0,n},I_{1,n},I_{2,n})$ is a deterministic three-way split with pairwise disjoint blocks and $$\left\lfloor \frac n4\right\rfloor\leq |I_{0,n}|,\qquad \left\lfloor \frac n4\right\rfloor\leq |I_{1,n}|,\qquad \left\lfloor \frac n4\right\rfloor\leq |I_{2,n}|.$$

Let $h_n$, the information-balance bandwidth for $\delta_n$, be as in Definition P-3, and let $$r_n=n^{-1/2}+\delta_n^{\kappa+1}h_n^\beta$$ be the frontier rate from Definition P-4. Then there exist constants $c,C\in\mathbb R$, with $0<c<C$, depending only on these regime constants, such that, for every threshold sequence and every deterministic split-block sequence satisfying the preceding conditions, all sufficiently large $n$ satisfy the following conclusions. The total-Gram estimator $\widehat\theta_n$ in Definition P-5, formed with split $B_n$, order $\ell=\lceil\beta\rceil-1$, bandwidth $h_n$, and threshold $\delta_n$, is observed-sample measurable. The corresponding bias-aware interval $\mathrm{CI}_n$ in Definition P-6, formed with the same split, order, bandwidth, and threshold, is an observed-sample measurable interval. For the causal minimax criteria $R_{n,\mathrm F}^\star$ and $L_{n,\mathrm F}^\star$ in Definition P-9, $$c r_n\leq R_{n,\mathrm F}^\star \leq \sup_{P^{\mathrm F}\in\mathcal M^{\mathrm F}} \mathbb E_{(P^{\mathrm F})^{\otimes n}} \left|\widehat\theta_n-\psi_{\delta_n}(P^{\mathrm F})\right| \leq C r_n,$$ $$\inf_{P^{\mathrm F}\in\mathcal M^{\mathrm F}} (P^{\mathrm F})^{\otimes n}\{\psi_{\delta_n}(P^{\mathrm F})\in \mathrm{CI}_n\} \geq 1-\alpha,$$ $$\sup_{P^{\mathrm F}\in\mathcal M^{\mathrm F}} \mathbb E_{(P^{\mathrm F})^{\otimes n}}\operatorname{len}(\mathrm{CI}_n) \leq C r_n.$$ Moreover, in the extended nonnegative-real order, after embedding the nonnegative real bounds into $\mathbb R_{\geq0}\cup\{\infty\}$, $$c r_n\leq L_{n,\mathrm F}^\star\leq C r_n .$$

Fix $J\in\mathbb N$ and constants $\kappa,c_-,c_+,p_{\min},\bar\delta,\alpha\in\mathbb R$. Suppose $$0<J, \qquad 0\leq \kappa, \qquad 0<c_-, \qquad c_-\leq \kappa+1\leq c_+, \qquad 0<p_{\min}\leq \frac{1}{J}, \qquad 0<\bar\delta<1, \qquad 0<\alpha<\frac12.$$ Then there exist constants $c,C\in\mathbb R$ with $0<c<C$ such that the following statements hold.

• (Uniform frontier bounds.) For every deterministic threshold sequence $(\delta_n)_{n\geq1}$ with $\delta_n\in[0,\bar\delta]$ for all $n$, and for every deterministic sequence $B_{\bullet}=(B_n)_{n\geq1}$ of admissible three-way split blocks, all sufficiently large $n$ satisfy the following bounds. With $$s_n=n^{-1/2}+\delta_n^{\kappa+1},$$ and with $R_{n,\mathrm{cont}}^\star$, $L_{n,\mathrm{cont}}^\star$, $R_{n,\mathrm{cont},\mathrm F}^\star$, and $L_{n,\mathrm{cont},\mathrm F}^\star$ as in Definition P-16, $$c s_n\leq R_{n,\mathrm{cont}}^\star \leq \sup_{P\in\mathcal M_{\mathrm{cont}}} \mathbb E_{P^{\otimes n}} \left|\widehat\theta_{n,\mathrm{cont}}-\theta_{\delta_n,\mathrm{cont}}(P)\right| \leq C s_n,$$ the interval $\mathrm{CI}_{n,\mathrm{cont}}$ has uniform coverage at least $1-\alpha$ over $\mathcal M_{\mathrm{cont}}$, its worst-case expected length is at most $C s_n$, and $$c s_n\leq L_{n,\mathrm{cont}}^\star.$$ Moreover, $$R_{n,\mathrm{cont},\mathrm F}^\star=R_{n,\mathrm{cont}}^\star, \qquad L_{n,\mathrm{cont},\mathrm F}^\star=L_{n,\mathrm{cont}}^\star,$$ and the same estimator and interval satisfy $$R_{n,\mathrm{cont},\mathrm F}^\star \leq \sup_{P^{\mathrm F}\in\mathcal M_{\mathrm{cont}}^{\mathrm F}} \mathbb E_{(P^{\mathrm F}_{\mathrm{obs}})^{\otimes n}} \left|\widehat\theta_{n,\mathrm{cont}}-\psi_{\delta_n}(P^{\mathrm F})\right| \leq C s_n,$$ with causal coverage at least $1-\alpha$ uniformly over $\mathcal M_{\mathrm{cont}}^{\mathrm F}$ and causal worst-case expected length at most $C s_n$.
• (Elbow and null-threshold rates.) For the continuity-only rate in Definition P-12, $$s_n\bigl(\delta_n=n^{-1/(2(\kappa+1))}\bigr) \asymp n^{-1/2}, \qquad s_n(0)\asymp n^{-1/2}.$$
• (Fixed positive thresholds.) For every deterministic threshold sequence $(\delta_n)_{n\geq1}$ with $\delta_n\in[0,\bar\delta]$ for all $n$, every $\delta_0\in(0,\bar\delta]$, and $\delta_n\to\delta_0$, $$R_{n,\mathrm{cont}}^\star \not\to 0, \qquad L_{n,\mathrm{cont}}^\star \not\to 0.$$