IID empirical bootstrap sample and mean statistic

Fix integers $d,n\ge 1$ and observed vectors $x_1,\dots ,x_n\in {\mathbb{R}}^d$, with the Borel sigma-algebra $\mathcal{B}({\mathbb{R}}^d)$ (The Borel sigma-algebra of a topological space). Their empirical law is the set function $P_n(A)=\frac{1}{n}{\sum }_{i=1}^{n}1\{x_i\in A\},A\in \mathcal{B}({\mathbb{R}}^d)\mathrm{.}$

Let ${\mathsf{I}}_n=\{\omega :[n]\to [n]\}$ be the finite set of index vectors. By The set $A^B$ of functions $B\to A$ between finite sets is finite, with $\mid A^B\mid =\mid A{\mid }^{\mid B\mid }$, it has $n^n$ elements. Give each index vector mass $n^{-n}$; the weights sum to one, so this is a finite probability space in the sense of Finite probability spaces, outcome weights, events, and event probabilities. For $j=1,\dots ,n$, let $I_j(\omega )=\omega (j)$ and put $X_j^{\ast }(\omega )=x_{I_j(\omega )}$. The ideal size-$n$ empirical bootstrap law conditional on the data is the pushforward of this finite uniform law under $\omega ⟼(x_{\omega (1)},\dots ,x_{\omega (n)})\mathrm{.}$ Thus conditional probabilities are finite averages over the $n^n$ index tuples. The empirical law is a probability measure, and the bootstrap coordinates are conditionally IID with that law, by the well-definedness results listed in justified_by.

For random observations $X_1,\dots ,X_n$, write ${\bar{X}}_n=n^{-1}{\sum }_i{X}_i$ coordinatewise, so each coordinate is the scalar sample mean in Partial sums, row sums and sample means. Write $P_n=n^{-1}{\sum }_i{\delta }_{X_i}$, using the Dirac measure The Dirac set function at a point, and let ${\bar{X}}_n^{\ast }$ be the bootstrap mean. If $g:{\mathbb{R}}^d\to \mathbb{R}$ is Borel (A measurable function between measurable spaces), then ${\widehat{\theta }}_n=g({\bar{X}}_n),{\widehat{\theta }}_n^{\ast }=g({\bar{X}}_n^{\ast })\mathrm{.}$ When $X_i$ are IID (Identical distribution and IID families) with finite second moment, write ${\mu }_j=\mathbb{E}{X}_{1j}$ and ${\mathrm{\Sigma }}_{jk}=\mathbb{E}[(X_{1j}-{\mu }_j)(X_{1k}-{\mu }_k)]$ (Moments, variance, and covariance on a probability space), and set $\theta =g(\mu )$. The functions of the data above are Borel statistics and estimators in the sense of Statistics and estimators; coordinatewise arithmetic preserves measurability (Arithmetic and lattice operations preserve measurability whenever they are defined). No bootstrap validity or interval coverage is part of this definition. For $n\ge 2$, the delete-one mean and plug-in statistic are ${\bar{X}}_{n,(-i)}=\frac{1}{n-1}{\sum }_{j\mathrm{\ne }i}{X}_j,{\widehat{\theta }}_{n,(-i)}=g({\bar{X}}_{n,(-i)})\mathrm{.}$