Four blocks, and only the ends matter

The weights a block carries, normalised so that the resample keeps the residuals' variance and only their covariances are attenuated. What separates these shapes, for everything that follows, is the value at the two ends and nothing about the middle: the first-order attenuation is −(w(0)² + w(1)²)/(2∫w²), which is -1.0000 for the rectangle, -0.3896 for the trapezoid cut off at half height, and exactly zero for both windows that reach the axis. The half-height trapezoid is in the table to be the case that separates a shape from a boundary value: it is smooth, it is tapered, and it buys none of the order the other two buy.

The bias in the long-run variance each construction implies, computed exactly on an autoregression at 0.7 with no sampling anywhere in it, against the block length. The rectangle's falls like ℓ^-1.00 and the geometric taper's like ℓ^-0.99; the two windows that reach zero at the block's ends fall like ℓ^-1.95 and ℓ^-1.97. The trapezoid cut off at half height falls like ℓ^-1.13, which is the first order with a small coefficient rather than the second order with a large one — its κ'(0) is -0.390 against the rectangle's -1.000, and at these block lengths the two terms are still competing.

Seven constructions on rows that repeat each other, at a block length of 5 and 200 draws, against the statistic's own 95% point of 3.0224 computed from three thousand draws of the same world. Reading down: a multiplier on every row keeps no dependence at all and is 44% short; a multiplier shared along a block keeps the triangle; a fixed block keeps the same triangle and is 8% closer, which is the pair that says a taper is not what decides this; the stationary bootstrap; the two tapered blocks, both further short than the untapered one at this block length; and errors generated from a fitted model, which is the only construction here not bounded by what the residuals report.