Delete-one jackknife acceleration

Input $n\ge 2$ real delete-one estimates ${\widehat{\theta }}_{(-1)},\dots ,{\widehat{\theta }}_{(-n)}$. Compute ${\bar{\theta }}_{(-)}=\frac{1}{n}{\sum }_{i=1}^n{\widehat{\theta }}_{(-i)},d_i={\bar{\theta }}_{(-)}-{\widehat{\theta }}_{(-i)},S_2={\sum }_i{d}_i^2\mathrm{.}$ If $S_2=0$, return undefined. Otherwise return the jackknife acceleration ${\widehat{a}}_J=\frac{{\sum }_i{d}_i^3}{6S_2^{3\mathrm{/}2}}\mathrm{.}$ The output is the finite-sample acceleration input used by the BCa procedure. For the special case ${\widehat{\theta }}_{(-i)}=g({\bar{X}}_{n,(-i)})$, Delete-one jackknife linearization for smooth functions of a mean proves first-order pseudovalue linearization and jackknife variance consistency under its full hypotheses: the $X_i\in {\mathbb{R}}^d$ are IID with $\mathbb{E}\mathrm{\parallel }{X}_1{\mathrm{\parallel }}^2<\mathrm{\infty }$; $g:{\mathbb{R}}^d\to \mathbb{R}$ is Borel and continuously Fréchet differentiable on an open neighborhood of $\mu =\mathbb{E}{X}_1$; and $0<a^T\mathrm{\Sigma }a<\mathrm{\infty }$ for $a=\mathrm{\nabla }g(\mu )$. This finite-sample algorithm and that first-order theorem make no BCa coverage claim.