Resampling schemes and the sampling-unit contract

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment (Statistical experiments) with a declared sampling or assignment mechanism. A sampling unit is one indexed unit that the declared mechanism treats as an indivisible draw or assignment object. Its unit record contains the complete observed data attached to that unit, including repeated measurements or labels that the analysis keeps together. Examples of sampling units are individuals, matched pairs, clusters, time blocks, and treatment-assignment units.

The declared mechanism also specifies a measurable unit-record space $(\mathcal{Z},\mathcal{F})$ and represents each unit record as an element of $\mathcal{Z}$. The record space may be the whole data space or one record-valued component of it; the convention is part of the experiment's sampling or assignment contract.

A resampling unit is the complete unit record or predeclared group of records that one draw of a resampling procedure selects. A procedure is unit-compatible for a stated target when it declares which records are sampled together and does not split a bundle whose joint sampling or assignment dependence is part of that target's data-generating design. Unit compatibility describes the procedure's input and dependence contract; it does not itself establish exact or asymptotic validity. A validity claim requires a separate theorem with hypotheses matched to the sampling or assignment law, target, and resample-size regime.

The conditional bootstrap law of observed units is a law on resampled data records given the fixed observations. When defined, a Bayesian posterior is instead a conditional law on parameter values under a specified prior and joint parameter-data model. They have different state spaces and are not identified by this definition. In particular, neither law determines the other.

Let $(\mathcal{Z},\mathcal{F})$ be a measurable unit space (Measurable spaces and measurable sets), let $n\ge 1$, and fix an observed ordered tuple $z_1,\dots ,z_n\in \mathcal{Z}$. Its empirical measure is the set function $P_n:=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{z_i},P_n(A)=\frac{1}{n}{\sum }_{i=1}^{n}1\{z_i\in A\}(A\in \mathcal{F}),$ where ${\delta }_{z_i}$ is the Dirac set function from The Dirac set function at a point. Repeated observations contribute once for each index at which they occur. The statement that $P_n$ is a probability measure is established in The empirical measure is a probability measure ↗.

For $n\ge 0$, let $[n]:=\{1,\dots ,n\}$, with $[0]=\mathrm{\varnothing }$. For an integer $r$ with $0\le r\le n$, let ${\mathcal{S}}_{n,r}:=\{I\subseteq [n]:\mathrm{\mid }I\mathrm{\mid }=r\}\mathrm{.}$ Then ${\mathcal{S}}_{n,r}$ is finite and nonempty, and assigning weight $1\mathrm{/}\mathrm{\mid }{\mathcal{S}}_{n,r}\mathrm{\mid }$ to each member defines a finite probability space on ${\mathcal{S}}_{n,r}$.

For every $n\ge 1$, every measurable space $(\mathcal{Z},\mathcal{F})$, and every observed tuple $z_1,\dots ,z_n\in \mathcal{Z}$, the empirical measure $P_n=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{z_i}$ from Empirical measure of observed units is a probability measure on $(\mathcal{Z},\mathcal{F})$.

Input. Let $n\ge 2$, let $(\mathcal{Z},\mathcal{F})$ be the declared measurable space of unit records, and fix observed records $z_1,\dots ,z_n\in \mathcal{Z}$ whose resampling unit satisfies Sampling units and the resampling-unit contract. For each tuple length $m$, declare a sample sigma-algebra ${\mathcal{F}}_m$ on the ordered tuple space ${\mathcal{Z}}^m$. Choose $1\le d<n$ and provide a statistic $T_{n-d}:({\mathcal{Z}}^{n-d},{\mathcal{F}}_{n-d})\to ({\mathbb{R}}^q,{\mathcal{B}}_q)$, where $q\ge 1$ is finite and ${\mathcal{B}}_q$ is the declared target sigma-algebra; "statistic" has the measurable-map meaning in Statistics and estimators.

Write $[n]:=\{1,\dots ,n\}$ for the indexing set; the finite subset law on these indices is established in Uniform laws on finite subset families.

Procedure. Form ${\mathcal{I}}_{n,d}:=\{I\subseteq [n]:\mathrm{\mid }I\mathrm{\mid }=d\}$. For every $I\in {\mathcal{I}}_{n,d}$, list the indices in $[n]\setminus I$ in increasing order and compute ${\widehat{\theta }}_{(-I)}:=T_{n-d}((z_i{)}_{i\in [n]\setminus I})\mathrm{.}$ Let $M_{n,d}:=\mathrm{\mid }{\mathcal{I}}_{n,d}\mathrm{\mid }$ and return the full collection of replicates together with the uniform average ${\bar{\theta }}_{(-d)}:=\frac{1}{M_{n,d}}{\sum }_{I\in {\mathcal{I}}_{n,d}}{\widehat{\theta }}_{(-I)}\mathrm{.}$ The finite, nonempty subset family and its uniform law are justified by Uniform laws on finite subset families.

This algorithm defines the replicates and their average only. Any variance or bias scaling is a separate statistic with an explicitly stated target; no scaling is implicit in the term delete-d jackknife, and the algorithm asserts no validity theorem.

Input. Let $n\ge 2$, let $(\mathcal{Z},\mathcal{F})$ be the declared measurable space of unit records, and fix observed records $z_1,\dots ,z_n\in \mathcal{Z}$ whose resampling unit satisfies Sampling units and the resampling-unit contract. Declare a sample sigma-algebra ${\mathcal{F}}_b$ on the ordered tuple space ${\mathcal{Z}}^b$, choose $1\le b<n$, and provide a statistic $T_b:({\mathcal{Z}}^b,{\mathcal{F}}_b)\to ({\mathbb{R}}^q,{\mathcal{B}}_q)$, where $q\ge 1$ is finite and ${\mathcal{B}}_q$ is the declared target sigma-algebra; "statistic" has the measurable-map meaning in Statistics and estimators.

Write $[n]:=\{1,\dots ,n\}$ for the indexing set; the finite subset law on these indices is established in Uniform laws on finite subset families.

Procedure. Draw $I$ uniformly from the finite family ${\mathcal{I}}_{n,b}:=\{J\subseteq [n]:\mathrm{\mid }J\mathrm{\mid }=b\}$, without replacement; list its indices in increasing order; and return the subsample $(z_i{)}_{i\in I}$ and statistic value $T_b((z_i{)}_{i\in I})$. The uniform law is well-defined by Uniform laws on finite subset families.

This is subsampling without replacement. It is different from drawing $b$ indices independently with replacement. The procedure defines a finite law only; validity requires a separate theorem for the data-generating design and asymptotic regime.

Input. Let $n\ge 1$, let $(\mathcal{Z},\mathcal{F})$ be the declared measurable space of sampling units, and fix the observed ordered tuple $z=(z_1,\dots ,z_n)$. Let $P_n$ be its empirical probability measure from Empirical measure of observed units and The empirical measure is a probability measure. Declare a sample sigma-algebra ${\mathcal{F}}_n$ on the ordered tuple space ${\mathcal{Z}}^n$ and provide a statistic $T_n:({\mathcal{Z}}^n,{\mathcal{F}}_n)\to ({\mathbb{R}}^q,{\mathcal{B}}_q)$, where $q\ge 1$ is finite and ${\mathcal{B}}_q$ is the declared target sigma-algebra; "statistic" has the measurable-map meaning in Statistics and estimators.

Write $[n]:=\{1,\dots ,n\}$ for the indexing set.

Procedure. Draw indices $I_1,\dots ,I_n$ independently and uniformly from $[n]$, so repeated indices are allowed. Set $Z_j^{\ast }:=z_{I_j}$ for $j=1,\dots ,n$. Return the bootstrap tuple $(Z_1^{\ast },\dots ,Z_n^{\ast })$ and the replicate $T_n(Z_1^{\ast },\dots ,Z_n^{\ast })$. Generate further replicates by repeating the index draws with fresh randomness. These are resampling units only when the tuple entries satisfy the unit contract in Sampling units and the resampling-unit contract.

The conditional IID law of the draws is proved in Empirical-law bootstrap draws are conditionally IID ↗. This algorithm does not claim that its conditional law consistently approximates any sampling law; such validity requires separate hypotheses and a separate theorem.

Fix $n\ge 1$, write $[n]:=\{1,\dots ,n\}$, let $(\mathcal{Z},\mathcal{F})$ be a measurable unit space, and fix an observed tuple $z=(z_1,\dots ,z_n)$. Let $P_n=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{z_i}$ be its empirical probability measure. For the bootstrap algorithm Size-n bootstrap from the empirical law, equip the conditional index vector $(I_1,\dots ,I_n)$ with the finite uniform law on ${\mathrm{\Omega }}_n:=\{f:[n]\to [n]\}$ and set $Z_j^{\ast }=z_{I_j}$. For every $A_1,\dots ,A_n\in \mathcal{F}$, ${\mathbb{P}}^{\ast }(Z_1^{\ast }\in A_1,\dots ,Z_n^{\ast }\in A_n)={\prod }_{j=1}^n{P}_n(A_j)\mathrm{.}$ Thus, conditional on the fixed observed tuple, $Z_1^{\ast },\dots ,Z_n^{\ast }$ are independent random elements with common law $P_n$ (Identical distribution and IID families).