Uniform subsampling coverage under root-law control

By [F2], under each fixed $P$ the IID observations have joint sampling law $P^{\otimes n}$. By [F5], $J_{n,P}$ is the CDF of the actual root $R_{n,P}$ under that law. Fix $\eta >0$. The two root-law limsup conditions imply that, for all sufficiently large $n$ and every $P\in \mathcal{P}$, both one-sided differences between $H_{m,P}$ and $J_{n,P}$ are at most $\eta \mathrm{/}2$. The feasible-root condition gives numbers ${\delta }_n\to 0$ such that, uniformly in $P$, the event $E_{n,P}:=\{{\sup }_x\mathrm{\mid }{\widehat{H}}_{m,n}(x)-H_{m,P}(x)\mathrm{\mid }\le \eta \mathrm{/}2\}$ has probability at least $1-{\delta }_n$. On this event, the triangle inequality gives ${\sup }_x\mathrm{\mid }{\widehat{H}}_{m,n}(x)-J_{n,P}(x)\mathrm{\mid }\le \eta$.

To see that neither root-law comparison follows from the m-regime, take $\mathcal{P}=\{{\delta }_0\}$, $\theta (P)=0$, $a_k=k$, and the measurable constant statistic $T_k={\mathbf{1}}_{\{k\text{ is odd}\}}\mathrm{/}k$. Let $m_n=\lfloor \sqrt{n}\rfloor$. Then $a_{m_n}\mathrm{/}{a}_n=m_n\mathrm{/}n\to 0$, $J_{n,P}={\delta }_{{\mathbf{1}}_{\{n\text{ is odd}\}}}$, and $H_{m_n,P}={\delta }_{{\mathbf{1}}_{\{m_n\text{ is odd}\}}}$. For every sufficiently large even $m$, the block $m^2\le n<(m+1{)}^2$ contains odd $n$, giving $H_{m_n,P}={\delta }_0$, $J_{n,P}={\delta }_1$, and ${\sup }_x(H_{m_n,P}(x)-J_{n,P}(x))=1$. For every sufficiently large odd $m$, the same block contains even $n$, giving $H_{m_n,P}={\delta }_1$, $J_{n,P}={\delta }_0$, and ${\sup }_x(J_{n,P}(x)-H_{m_n,P}(x))=1$. Each root-law comparison can therefore fail under the stated m-regime, even with $a_m\mathrm{/}{a}_n\to 0$.

Apply [F4], part 3, with $F=J_{n,P}$, $\widehat{G}={\widehat{H}}_{m,n}$, CDF error $2\eta$, and exceptional-event probability ${\delta }_n$. Step 1.1 supplies the event on which the CDF error is at most $\eta$, so the lemma's pointwise tail inclusions bound the joint miss event without assuming independence of $R_{n,P}$ and that event. The root interval $[q_{{\widehat{H}}_{m,n}}({\alpha }_1),q_{{\widehat{H}}_{m,n}}(1-{\alpha }_2)]$ therefore misses $R_{n,P}$ with probability at most ${\alpha }_1+{\alpha }_2+2\eta +{\delta }_n$, uniformly in $P$.

Suppose only the first root-law limsup condition and the feasible-root condition hold. For any $\eta >0$, all sufficiently large $n$ satisfy $H_{m,P}(x)-J_{n,P}(x)\le \eta \mathrm{/}2$ uniformly in $P,x$, and on an event $E_{n,P}$ of probability at least $1-{\delta }_n$ we have ${\widehat{H}}_{m,n}(x)-H_{m,P}(x)\le \eta \mathrm{/}2$ for all $x$. Hence on $E_{n,P}$, ${\widehat{H}}_{m,n}-J_{n,P}\le \eta$. If ${\alpha }_2+\eta <1$, quantile ordering in [F4] gives $q_{J_{n,P}}(1-{\alpha }_2-\eta )\le q_{{\widehat{H}}_{m,n}}(1-{\alpha }_2)$. Thus the joint failure event $\{R_{n,P}>q_{{\widehat{H}}_{m,n}}(1-{\alpha }_2)\}\cap E_{n,P}$ is contained in $\{R_{n,P}>q_{J_{n,P}}(1-{\alpha }_2-\eta )\}$, whose probability is at most ${\alpha }_2+\eta$ by [F4]; adding $P(E_{n,P}^c)\le {\delta }_n$ bounds total failure. Letting $n\to \mathrm{\infty }$ and then $\eta \downarrow 0$ proves uniform asymptotic coverage at least $1-{\alpha }_2$ for the lower confidence bound; if ${\alpha }_2+\eta \ge 1$, the intermediate lower bound is trivial. For the second assertion, if only the second root-law condition holds, then on the same event $J_{n,P}-{\widehat{H}}_{m,n}\le \eta$. When ${\alpha }_1+\eta <1$, [F4] gives $q_{{\widehat{H}}_{m,n}}({\alpha }_1)\le q_{J_{n,P}}({\alpha }_1+\eta )$. The joint failure event $\{R_{n,P}<q_{{\widehat{H}}_{m,n}}({\alpha }_1)\}\cap E_{n,P}$ is contained in $\{R_{n,P}<q_{J_{n,P}}({\alpha }_1+\eta )\}$, whose probability is at most ${\alpha }_1+\eta$ by [F4]; adding the complement probability and letting $\eta \downarrow 0$ proves uniform asymptotic coverage at least $1-{\alpha }_1$ for the upper confidence bound. If ${\alpha }_1+\eta \ge 1$, the intermediate lower bound is trivial.

By the definition of ${\mathcal{C}}_n$, $P\in {\mathcal{C}}_n$ exactly when $\theta (P)\in I_n$. Since $a_n>0$, this is algebraically equivalent to $q_{\widehat{H}}({\alpha }_1)\le a_n({\widehat{\theta }}_n-\theta (P))\le q_{\widehat{H}}(1-{\alpha }_2)$. It is the coverage event in [F3]. Thus its uniform lower bound is $1-{\alpha }_1-{\alpha }_2-2\eta -{\delta }_n$. Take the liminf in $n$, then let $\eta \downarrow 0$, to obtain the stated inequality. The proof uses no independence between the two tail events.

To see that feasible-root control can fail even when both root-law comparisons hold exactly, take $\mathcal{P}=\{P\}$ with $P={\delta }_0$, $\theta (P)=0$, $T_k(z_1,\dots ,z_k)=1\mathrm{/}k$, and $a_k=k$, again with $m_n=\lfloor \sqrt{n}\rfloor$. For all $n\ge 4$, $1\le m_n<n$, $m_n\to \mathrm{\infty }$, $m_n\mathrm{/}n\to 0$, and $a_{m_n}\mathrm{/}{a}_n=m_n\mathrm{/}n\to 0$. The actual size-$n$ and size-$m_n$ roots are both identically $1$, so $J_{n,P}=H_{m_n,P}$. For either resampling method, every resampled statistic is $1\mathrm{/}{m}_n$, hence ${\widehat{H}}_{m,n}={\delta }_{1-m_n\mathrm{/}n}$. Its sup-CDF distance from $H_{m_n,P}={\delta }_1$ is $1$, so feasible-root control fails. With ${\alpha }_1={\alpha }_2=1\mathrm{/}4$, both quantiles equal $1-m_n\mathrm{/}n$ and $I_n=\{m_n\mathrm{/}{n}^2\}$, which excludes $\theta (P)=0$; coverage is zero for every $n\ge 4$.