Moving block bootstrap

Input: observations $x_1,\dots ,x_n$ in a real Hilbert space $H$, and an integer block length $b$ with $1\le b\le n$.

Output: a pseudo-series $X_1^{\ast },\dots ,X_n^{\ast }$, generated by Overlapping block-index resampling: within each selected block retain the original order and concatenate blocks in their draw order, truncating only the last block when $b\nmid n$. The conditional bootstrap sample mean is ${\bar{X}}_n^{\ast }=n^{-1}{\sum }_{t=1}^n{X}_t^{\ast }$. Its conditional center is $E^{\ast }{\bar{X}}_n^{\ast }$, where $E^{\ast }$ averages over the independent uniform block starts while the observations remain fixed.

This algorithm specifies the resampling law only. It does not assert that the conditional law approximates the sampling law for any data-generating process or block-length regime.

Under Sampling units and the resampling-unit contract, take each observation $x_t$ to be a complete time-indexed unit record. Each draw selects an ordered group of $b$ consecutive records; the output retains every record of each complete block and only the initial $r=n-b\lfloor n\mathrm{/}b\rfloor$ records of the final block when $r>0$. Thus selected length-$b$ groups are not declared indivisible sampling units. Unit compatibility for a stated target requires that block boundaries and final truncation do not split any bundle declared indivisible by that target's sampling or assignment design. This is the sampling-unit interface, not a bootstrap-validity result.