Normal Distribution
nothing is called 95% until it has been counted
Concept

Block count — where it appears

The number of whole blocks a series of n rows is cut into at block length ℓ, the integer part of n over ℓ. A variance estimated from the block means has one fewer degrees of freedom than there are blocks, and that count prices every interval built on it before anything is resampled.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

One t interval with its scale read three ways, 120 rows in blocks of 32. 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.

A block at every starting row

The t interval on the block count reads its scale from the disjoint blocks, three of them when 120 rows are cut into blocks of thirty-two, and throws away the other eighty-six blocks the series contains. Reading a block at every starting row leaves the scale's attenuation where it was — the two expectations agree to within a hundredth at every cell — and nearly doubles the degrees of freedom it realises, so the interval loses a third of its width at the same coverage. Where the interval falls short, at short blocks, it falls short by the same amount however the blocks are read.

taper · 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 Coverage against block length, 120 rows. 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%.

A length chosen for the interval

The squared-error rule for a block length weighs the scale's bias against its variance. Rebuilt for a block at every starting row, whose variance is two thirds as large, it asks for blocks fourteen per cent longer and buys four tenths of a point of coverage. A t interval has already paid for the scale's variance in its multiplier, so its own coverage is predicted by the bias alone — within seven tenths of a point up to blocks of twenty-eight rows — and a rule that chooses the shortest length that prediction says will cover asks for blocks two and a half to three times longer and lands within about a point of 95% at 120, 240 and 480 rows. The price is width, and it is paid where the series is short.

taper · 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

Named alongside it

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

Block lengthClosed formConfidence intervalCoverageDegrees of freedomDependenceInterval widthLong-run varianceBlock bootstrapBias-varianceEstimated variancePercentile interval

All concepts