Delete-d jackknife replicates and their uniform average

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.