Second-Order Risk in Multi-Arm Randomization

The unrestricted labeled-schedule minimax risk for contrast $c$ is $$\rho_n(c)=\inf_{\mathcal D\in\Delta(\mathcal A_K^n),\,\widehat\tau}\max_{z\in\mathcal T_K^n}\mathbb E_{\mathcal D}\left[\left\{\widehat\tau((A_i,Y_i^{\mathrm{obs}})_{i=1}^n)-\tau_c(z)\right\}^2\right].$$

Fix $n\in\mathbb N$, $K\ge 2$, and a nonzero zero-sum contrast $c$ in the labeled-schedule and finite-orbit games of Definition P-1, Definition P-3. Then:

• (Lossless symmetrization.) For every labeled procedure $p$, there are an invariant labeled procedure $\bar p$ and an orbit procedure $q$ such that $\bar p$ is the common-unit-permutation average of $p$, and for every complete labeled schedule $z\in\mathcal T_K^n$, $$R_n(\bar p;z) \le \frac{1}{|\mathfrak S_n|} \sum_{\sigma\in\mathfrak S_n} R_n(p;\sigma z).$$ The orbit procedure $q$ is dominated by this averaged invariant representative in the corresponding orbit risk.
• (Exact invariant correspondence.) Invariant labeled procedures are in bijection with orbit procedures $(\pi,\delta)$. Under this bijection, each invariant labeled procedure and its orbit representative realize the same allocation-count and observation-count rule, and for every $z\in\mathcal T_K^n$, $$R_n(q;\operatorname{counts}(z))=R_n(\bar p;z).$$
• (Target preservation.) If $m=\operatorname{counts}(z)$, with $m_t=\#\{i:t_i=t\}$, then $$\tau_c(z) = \frac{1}{n}\sum_i\sum_{a\in\mathcal A_K} c_a\,t_{i,a} = \frac{1}{n}\sum_t m_t\sum_{a\in\mathcal A_K} c_a\,t_a = \tau_c(m).$$
• (Value equality.) The unrestricted labeled-schedule minimax risk equals the finite response-type orbit minimax value: $$\rho_n(c)=G_n(c).$$
• (Orbit saddle point.) There exist an orbit procedure $q^\star$ and a prior $\nu$ on $\mathcal M_{n,K}$ such that, for every response-count vector $m$ and every orbit procedure $q$, $$R_n(q^\star;m)\le G_n(c) \qquad\text{and}\qquad G_n(c)\le \mathbb E_{\nu}\!\left[R_n(q;m)\right].$$

Let $$S_c=\{a\in\mathcal A_K:c_a\ne 0\},\qquad L_c=\sum_{a\in\mathcal A_K}|c_a|,\qquad q_a^\star=\frac{|c_a|}{L_c}\quad(a\in S_c).$$ Assign each $A_i$ independently on $S_c$ with $\Pr(A_i=a)=q_a^\star$. The projected contrast-weighted Horvitz--Thompson rule is $$\widehat\tau_{\mathrm{cHT}}^\star = \operatorname{proj}_{[-L_c/2,L_c/2]}\!\left[ n^{-1}\sum_{i=1}^n\sum_{a\in S_c} c_a\mathbf 1\{A_i=a\}\frac{Y_i^{\mathrm{obs}}-1/2}{q_a^\star} \right].$$

Fix a natural number $K$ and a nonzero zero-sum contrast $c$ on the treatment-arm set $\mathcal A_K$, in the labeled-schedule game of Definition P-1. Let $L_c$, $q_a^\star$, and $\widehat\tau_{\mathrm{cHT}}^\star$ be the contrast norm, contrast-weighted assignment probabilities, and projected contrast-weighted Horvitz--Thompson rule from Definition P-4, and set $$C_0(c)=\frac{L_c^2}{4}.$$ Then the unrestricted labeled-schedule minimax risk satisfies $$\lim_{n\to\infty} n\,\rho_n(c)=C_0(c).$$ Moreover, for every $n\ge 1$, the independent contrast-weighted design together with $\widehat\tau_{\mathrm{cHT}}^\star$ attains the finite-sample worst-case bound $$\max_{z\in\mathcal T_K^n} R_n(\mathcal D^\star,\widehat\tau_{\mathrm{cHT}}^\star;z) \le \frac{C_0(c)}{n}.$$

For every fixed $K\geq 2$ and every nonzero zero-sum contrast $c$, let $\lambda_c$, the smallest nonzero normalized response-type contrast magnitude, be $$\lambda_c = \min\left\{ \frac{\left|\sum_{a\in\mathcal A_K} c_a t_a\right|}{L_c/2} : t\in\mathcal T_K,\ \sum_{a\in\mathcal A_K} c_a t_a\neq 0 \right\}.$$ Define $$C_0(c)=\frac{L_c^2}{4}, \qquad A_c=\frac{\sqrt{\lambda_c}}{2}-\frac{\lambda_c}{16}, \qquad B_c=\frac{7\lambda_c}{16}, \qquad \kappa_c=\frac{1}{2}\min\{A_c,B_c\},$$ and $$d_n(c)=\frac{C_0(c)}{n}-\rho_n(c), \qquad s_n=n^{4/3}.$$ The following assertions hold:

• (Spacing.) The constants satisfy $$0<\lambda_c\leq 1, \qquad 0<\kappa_c.$$
• (First-order envelope.) For every $n$, $$\rho_n(c)\leq \frac{C_0(c)}{n}.$$
• (Uniform shrinkage bound.) There is a finite integer $N_c$ such that, for every integer $m\geq N_c$, $$4\sqrt{\lambda_c}\,m\exp\{-m^{1/3}/8\} \leq \frac{\sqrt{\lambda_c}}{2}-\frac{\lambda_c}{16},$$ and, for every $n\geq N_c$, the explicit clipped-shrinkage procedure $\widehat\tau_n^{\mathrm{sh}}$ satisfies $$\sup_{z\in\mathcal T_K^n} R_n(\mathcal D^\star,\widehat\tau_n^{\mathrm{sh}};z) \leq C_0(c)\left\{n^{-1}-\kappa_c n^{-4/3}\right\}.$$
• (Second-order envelope.) The normalized second-order improvement satisfies $$C_0(c)\kappa_c \leq \liminf_{n\to\infty} n^{4/3}d_n(c) \leq \limsup_{n\to\infty} n^{4/3}d_n(c) \leq 43C_0(c).$$
• (Regular-variation index.) For every positive sequence $a_n$, every $\beta$, and every $C>0$, if $a_n$ is regularly varying with index $\beta$ and $$a_n d_n(c)\to C,$$ then $\beta=4/3$.

For $K\geq 2$, $n,M\geq 1$, and a rational nonzero zero-sum contrast $c\in\mathbb Q^K$, define the rational contrast grid-action value $\Lambda_{n,M}(c)$ as the optimal value of the following finite linear program. Let $$h_c=L_c/2, \qquad \Gamma_{M,c}=\{-h_c+jh_c/M:j=0,\ldots,2M\}.$$ The program minimizes $u$ over variables $\pi_r\geq0$ and $w_{r,x,g}\geq0$, subject to $$\sum_{r\in\mathcal R_{n,K}}\pi_r=1,$$ $$\sum_{g\in\Gamma_{M,c}}w_{r,x,g}=\pi_r \quad\text{for every }(r,x),$$ and $$\sum_{r\in\mathcal R_{n,K}}\sum_x\sum_{g\in\Gamma_{M,c}} P_m(x\mid r)w_{r,x,g}\{g-\tau_c(m)\}^2\leq u \quad\text{for every }m\in\mathcal M_{n,K}.$$

Fix $K\ge 2$ and a nonzero zero-sum real contrast $c$ on $\mathcal A_K$. For real contrasts $c_1,c_2$, write $$\eta(c_1,c_2)=\frac12\sum_{a\in\mathcal A_K}|c_1(a)-c_2(a)|, \qquad C_0(c_1)=\frac{L_{c_1}^2}{4}.$$ A tuple $(B,U,R^-,R^+)$ is a transferred certificate for $(n,M,c,q)$, where $q\in\mathbb Q^K\setminus\{0\}$ is zero-sum, if there exist an invariant allocation-count distribution $\pi$, a rational grid-action weight $w$, a rational epigraph value $u$, a rational response-count prior $\nu$, an invariant barycenter rule $\delta$, a projected upper procedure, and a schedule prior such that the exact rational primal-dual grid certificate and barycenter identities hold for $q,\pi,w,u,\nu,\delta$, and $$B=B_{n,q}(\nu),\qquad U=U_{n,M}(q;\pi,\delta),\qquad u=\Lambda_{n,M}(q),$$ $$B\le \rho_n(q)\le U\le u \le B+\frac{C_0(q)}{4M^2},$$ $$R^-=\left\{\max\bigl(0,\sqrt B-\eta(c,q)\bigr)\right\}^2, \qquad R^+=\left(\sqrt U+\eta(c,q)\right)^2,$$ $$R^-\le \rho_n(c)\le R^+,$$ and $$R^+-R^-\le \frac{5C_0(q)}{4M^2} +4\sqrt{C_0(q)}\,\eta(c,q)\,n^{-1/2} +3\eta(c,q)^2.$$ The projected upper procedure certifies worst-case risk at most $R^+$, and the schedule prior certifies Bayes risk at least $R^-$. Then the following hold.

• (Finite-grid transfer.) For every rational zero-sum contrast $q\in\mathbb Q^K\setminus\{0\}$ and every pair of integers $n,M\ge 1$, there exist $B,U,R^-,R^+\in\mathbb R$ forming a transferred certificate for $(n,M,c,q)$.
• (Real-contrast approximation.) There exist rational zero-sum contrasts $c^{(n)}\in\mathbb Q^K\setminus\{0\}$ and real sequences $B_n,U_n,R_n^-,R_n^+$ such that, for every arm $a\in\mathcal A_K$, $$c^{(n)}(a)\longrightarrow c(a),$$ $$n^{5/6}\eta(c,c^{(n)})\longrightarrow 0,$$ for every $n\ge 1$, $(B_n,U_n,R_n^-,R_n^+)$ is a transferred certificate for $(n,n,c,c^{(n)})$, and $$n^{4/3}\bigl(R_n^+-R_n^-\bigr)\longrightarrow 0.$$
• (Exact rational case.) If a rational zero-sum contrast $q\in\mathbb Q^K\setminus\{0\}$ embeds coefficientwise as $c$, then for every $n\ge 1$ there exist $B,U,R^-,R^+\in\mathbb R$ forming a transferred certificate for $(n,n,c,q)$ and satisfying $$U-B\le \frac{C_0(c)}{4n^2}.$$

For $K=3$, let $c^\dagger=(1,-1/2,-1/2)$, and let $\mathcal D^\star$ be the independent design assigning each unit with probabilities $q^\star=(1/2,1/4,1/4)$. For any $n\geq 1$, any schedule $z\in\mathcal T_3^n$, and any unit $i$, define $$Z_i=\operatorname{sign}(c^\dagger_{A_i})(2Y_i^{\mathrm{obs}}-1), \qquad \mu_i=t_{i,1}-\frac{t_{i,2}+t_{i,3}}{2}.$$ Then the following statements hold.

• (Score moments.) For every $n\geq 1$, every $z\in\mathcal T_3^n$, and every unit $i$, $$\mathbb E_{\mathcal D^\star}(Z_i\mid z)=\mu_i, \qquad \operatorname{Var}_{\mathcal D^\star}(Z_i\mid z)=1-\mu_i^2 .$$
• (Average-score risk.) For every $n\geq 1$ and every $z\in\mathcal T_3^n$, $$\mathbb E_{\mathcal D^\star}\!\left[ \left\{\frac{1}{n}\sum_i Z_i-\tau_{c^\dagger}(z)\right\}^2 \,\middle|\, z\right] = \frac{1}{n}-\frac{1}{n^2}\sum_i \mu_i^2 .$$
• (Scalar minimax value.) For the scalar experiment at $n=3$ that retains $(Z_1,Z_2,Z_3)$ and lets each $\mu_i$ range over the five feasible values, its minimax value is $$V_3^Z=1-\frac{\sqrt 3}{2}.$$
• (Full-data certificate.) There exists a clipped full-data estimator using the observed arm labels and outcomes under $\mathcal D^\star$ whose worst-case risk over $z\in\mathcal T_3^3$ is $$\frac{511653}{4000000}.$$ Moreover, $$\frac{511653}{4000000} < \frac{18213}{136000} < 1-\frac{\sqrt 3}{2}.$$
• (Scalar moment envelope.) For every $n$, every real $M$, and every feasible three-arm scalar total $M=|\sum_i\mu_i|$, the minimum attainable value of $\sum_i\mu_i^2$ among scalar mean vectors with $|\sum_i\mu_i|=M$ is $$\begin{cases} M/2, & M\leq n/2,\\ (3M-n)/2, & M>n/2. \end{cases}$$

Thus the scalar signed-score experiment has exact value $V_3^Z$ at $n=3$, while the full observed arm-label-and-outcome experiment under the same independent $q^\star$ design attains the smaller certified worst-case risk displayed above.