Three intervals, one shortfall

What each of three intervals actually covers, at four rules and two block windows, over 300 samples of 120 rows. All three are built from the same resamples on the same draws, so a difference between them is a difference in what is done with the resampled series. Not one of the twenty-four cells reaches the ninety-five per cent it promises. The studentised interval runs from 75.7% to 92.3%, the percentile interval — the earlier field's — from 80.0% to 89.7%, and a normal interval on the same scale from 81.7% to 89.0%. The standard repair for a percentile interval's shortfall does not repair it.

The margin between the two block windows in points of coverage, under each of four rules, on each of three intervals built from the same resamples, over 300 draws. Positive is the tapered window covering better. On the percentile interval the taper wins at all four rules, by 5.33, 1.67, 5.00 and 4.00 points, which is the earlier field's own reading. On the studentised interval the rectangle wins at all four, by 4.33, 7.00, 4.33 and 2.67. And a normal interval, which uses no resampling at all, puts the two within a third of a point at every rule — so the disagreement is manufactured entirely by what is done with the resamples.

How much wider the studentised interval is than the percentile one, at every sample size and every block length on the grid, with the number of whole blocks each cell leaves written beneath. Read across a row and the block length changes; read down a column and the sample size does. The penalty is nearly a function of the block count alone: the cells at 15 blocks read 1.16, 1.20, 1.17, 1.15, 1.13, 1.10 across three sample sizes and three block lengths, while the cells at one block length read anything from 1.10 to 3.95. The largest penalty on the grid is 3.95, at the cell with 3 whole blocks in it.

The coverage of a t interval on the block count against the studentised bootstrap interval's, at all 24 cells of the grid — three sample sizes, four block lengths, two windows — on the same 240 draws a cell. No point is below the diagonal: the t interval covers more often at every one of them, by 0.42 to 10.42 points and 4.11 on average. It reaches the 95% line at 2 cells, 15 blocks of 32 at 480 rows under both windows, where no resampled interval on the grid reaches it at all. The t interval resamples nothing, so under the two windows it is the same interval and each of its values appears twice, against two different studentised ones.

What a 95% t interval for the mean of a first-order autoregression at 0.7 covers, and how wide it is, when its block-means scale is read from the 3 disjoint blocks of 32 rows, from a block at every one of the 89 starting rows, and from those overlapping blocks tapered — over 8,000 draws. Disjoint: 94.4% at a width of 1.560, with a multiplier of 4.30. Every starting row: 94.8% at 1.054, multiplier 2.74. Tapered: 95.0% at 1.020, multiplier 2.57.

The coverage of the t interval for the mean of a first-order autoregression at 0.7 on 120 rows, at every block length on a fine grid, with its scale read from the disjoint blocks and from a block at every starting row, over 8,000 draws. The squared-error length for disjoint blocks at the true persistence is 13.9 rows, and the same algebra for the overlapping scale gives 16.0, marked. The overlapping interval first reaches 95% — to within a tenth of a point — at blocks of 32 rows, where it covers 94.9% at a width of 1.056; at the overlapping squared-error length it covers about 92.5%.

For a five-period moving average at 120 rows, the share of the mean's variance the overlapping block-means scale expects at each block length, computed exactly from the law's covariance, beside the share two rules believe on average over 4,000 draws: the share a first-order autoregression at the sample's lag-one autocorrelation implies, and the share computed from the sample's own autocovariances to the cut-off lag the sample chooses. At blocks of 24 rows the law's share is 0.93; the geometric rule believes 0.78 and the sample-read rule 0.94.

Coverage at 120 rows when the block length is chosen from a share read off the sample's autocovariances to a stated lag, as read and corrected for the centring. first-order autoregression, as read: 93.9% at lag 6, 93.3% at lag 12, 88.7% at lag 24, 84.5% at lag 40. the same, corrected: 94.0% at lag 6, 94.5% at lag 12, 91.9% at lag 24, 89.2% at lag 40. moving average, as read: 94.5% at lag 6, 92.3% at lag 12, 88.7% at lag 24, 85.0% at lag 40. the same, corrected: 94.7% at lag 6, 93.7% at lag 12, 91.7% at lag 24, 89.8% at lag 40. a break, as read: 92.1% at lag 6, 93.1% at lag 12, 90.1% at lag 24, 83.5% at lag 40. the same, corrected: 92.2% at lag 6, 93.8% at lag 12, 92.2% at lag 24, 89.7% at lag 40.