m-out-of-n bootstrap and subsampling

Let $n\ge 1$ be an integer, let $(\mathcal{Z},\mathcal{C})$ be a measurable observation space, let $P$ be a probability law on it, and let $Z_1,\dots ,Z_n$ be IID $P$-distributed sampling units (Identical distribution and IID families). For each $1\le k\le n$, let $T_k$ be a measurable real-valued statistic and write ${\widehat{\theta }}_k=T_k(Z_1,\dots ,Z_k)$; the target is a scalar $\theta (P)$ and $a_k>0$ is its declared normalization at sample size $k$. The empirical law $P_n:=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{Z_i}$ is a probability law by A finite empirical measure is a probability law.

For an integer $1\le m\le n$, the m-out-of-n bootstrap draws $Z_1^{\ast },\dots ,Z_m^{\ast }$ conditionally IID from $P_n$ and computes $R_{m,n}^{\ast }:=a_m(T_m(Z_1^{\ast },\dots ,Z_m^{\ast })-{\widehat{\theta }}_n)\mathrm{.}$ The conditional distribution of this root given the observed sample is the bootstrap approximation. The ordinary n-out-of-n bootstrap is the case $m=n$ and uses $a_n$.

For $1\le m<n$, subsampling without replacement chooses a subset $I=\{i_1<\cdots <i_m\}$ uniformly from the $\left(\genfrac{}{}{0px}{}{n}{m}\right)$ subsets of $\{1,\dots ,n\}$ and computes $R_{m,n}^{\mathrm{s}\mathrm{u}\mathrm{b}}:=a_m(T_m(Z_{i_1},\dots ,Z_{i_m})-{\widehat{\theta }}_n)\mathrm{.}$ Its conditional law is the uniform distribution over those subsets. With replacement and without replacement are distinct conditional laws, even though both use the same declared statistic $T_m$.

The asymptotic regime considered on this page has integer $m=m_n\to \mathrm{\infty }$ and $m_n\mathrm{/}n\to 0$. The resampled root uses the actual size-$m$ scale $a_m$; validity does not follow merely from this regime. For inference based on a centered root, the scale ratio $a_m\mathrm{/}{a}_n\to 0$ is an additional root-specific centering condition when it is used. A design with dependent sampling units requires a separate design-matched resampling procedure. This definition does not identify a bootstrap law with a Bayesian posterior or a design-randomization law. Extrapolation from several resample sizes is not asserted here; it requires a separate proof of the source-specific regular expansion and remainder regime.