Concept

Percentile interval — where it appears

An interval read straight off the quantiles of a resampled statistic, with no scale estimated and no correction applied. It is the cheapest bootstrap interval and it inherits both the resampled distribution's skewness and its scale error, which is the shortfall a studentised interval is meant to repair.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

An interval that carries its scale

A percentile interval inherits the resampled distribution's skewness and its scale error together. The standard repair is one extra variance per resample. It was named and not run, so this runs it.

student · Bootstrap

What studentising costs

Averaged over eight cells the studentised interval is 2.09 times as wide as the percentile one and covers 0.46 points better. At the block lengths the rules choose, the scale it divides by rests on two or three numbers.

student · Bootstrap

Four rules of four change sign. 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.

The ordering reverses again

One field found two of four rules changing sign between two readings of one resampling. Turn the same resamples into a studentised interval instead of a percentile one and all four change sign.

student · Bootstrap

One penalty, read along two dials. 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 count or the length

A block length and a block count are one number read two ways at one sample size. Read at three, the studentised interval's width penalty tracks the count — with an R² of 0.9911 against a closed form that has no length in it — and its coverage tracks the length.

student · Bootstrap

Four intervals at 3 blocks of 32 rows. What four 95% intervals for the mean of a first-order autoregression at 0.7 cover, and how wide they are on average, at 120 rows cut into 3 whole blocks of 32, over 240 draws with 200 resamples each under the rectangle. The normal interval, the block-means variance with 1.96, covers 82.1% at a width of 0.708. The percentile interval covers 82.1% at 0.632 and the studentised one 90.4% at 2.495. The fourth resamples nothing: it is the normal interval with 1.96 replaced by Student's t on 2 degrees of freedom, and it covers 94.2% at 1.555, 0.62 times the studentised interval's width.

The interval with no resampling in it

Replace 1.96 in a normal interval on the block-means variance with Student's t on one fewer degrees of freedom than there are whole blocks, and resample nothing. Across twenty-four cells it covers at least as often as the studentised bootstrap interval at every one, by 0.42 to 10.42 points; it is narrower wherever seven blocks or fewer are left; and at fifteen blocks of 32 it covers 95.0%, which no resampled interval on the grid reaches.

student · Bootstrap

The two standardised means, and the shape of Fieller's set for their ratio at a = 3, d = 1. Each dot is one pair (zx, zy) drawn around (3, 1). Outside the horizontal band |zy| > 1.96 Fieller's set is a bounded interval, with probability 17.01%; inside the band and outside the disc of radius 1.96 it is everything outside an interval, 75.03%; inside the disc it is the whole line, 7.96%. On 40,000 counted draws Fieller covers ρ = 3.00 95.21% of the time and the delta interval 82.48%.

A ratio whose interval has to be the whole line

The delta interval for a ratio of two means covers 95.61% when the denominator is eight standard errors from zero and 1.10% at a ten-thousandth of one, and ten times as wide it still covers only 3.48%. Linearising is not the fault. Gleser and Hwang proved that every interval that is always finite fails the same way, so an interval that keeps its promise has to be the whole line some of the time.

normal · Clt

Named alongside it

The objects these essays reach for when they reach for this one.

Confidence interval Coverage Block bootstrap Block length Dependence Estimated variance Long-run variance Resampling Studentised bootstrap Closed form Interval width Monte Carlo

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