Tapered block bootstrap

Input: observations $Y_1,\dots ,Y_n$ in a real Hilbert space $H$, an integer $1\le b\le n$, and a window $w:\mathbb{R}\to [0,1]$ that vanishes off $[0,1]$, is positive on a neighbourhood of $1\mathrm{/}2$, satisfies $w(1-u)=w(u)$, and is nondecreasing on $[0,1\mathrm{/}2]$. Put $w_b(r)=w((r-\frac{1}{2})\mathrm{/}b)$ for $1\le r\le b$, $W_b=({\sum }_{r=1}^b{w}_b(r{)}^2{)}^{1\mathrm{/}2}$, and $a_b=\sqrt{b}\mathrm{/}{W}_b$. The procedure is admissible only when $W_b>0$.

Center and taper: set ${\bar{Y}}_n=n^{-1}{\sum }_{t=1}^n{Y}_t$ and $R_t=Y_t-{\bar{Y}}_n$. For each start $j\in \{1,\dots ,n-b+1\}$, form the block

$(a_b{w}_b(1)R_j,\dots ,a_b{w}_b(b)R_{j+b-1})\mathrm{.}$

Resample: use the independent uniform block starts from Overlapping block-index resampling, multiply each selected centered observation by its within-block taper and the factor $a_b$, concatenate in draw order, and retain the first $n$ entries. The conditional bootstrap mean is centered by its own conditional expectation when used to approximate the sampling law of $\sqrt{n}{\bar{Y}}_n$.

The factor $a_b$ compensates for the window's squared-weight sum in the block-sum variance calculation; this definition alone gives no bootstrap validity result.

For a dependent time series, the selected length-$b$ consecutive run is the resampling unit under Sampling units and the resampling-unit contract. The taper changes each unit record's within-block weight but the whole block is selected and resampled together; validity remains a separate theorem.