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

Plug in — where it appears

A quantity computed from the data and substituted into a formula that was derived as though it were known. The substitution is what makes an unknown nuisance affordable and it is not free, because nothing in the formula accounts for the value having been estimated rather than given.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Three rules and a target none of them is aimed at. Which block length each rule picks, over 400 samples of 120 rows, for the tapered window. Two of the rules are points: a length written into a protocol is 8.00 on every draw and the rule of thumb is 4.00, because n to the one third does not read the data at all. The plug-in reads the sample's own persistence and lands at 14.36 with a standard deviation of 2.93. The length that would actually have been best on that draw averages 24.57 with a standard deviation of 16.23 and runs from 10 to 48 between its tenth and ninetieth percentiles. The target moves five times as much as the best estimate of it does, which is why no rule can be close to it and why the two that do not try are not merely worse — they are somewhere else.

The length nobody has

Every comparison of block windows in this collection is made at each window's own best block length. That length has a standard deviation of sixteen across draws and averages twenty-five. No rule is aimed at it.

feasible · Bootstrap Which window is better depends on who chose the block length. The margin between a rectangular block and a tapered one, on 400 samples of 120 rows, under four rules for choosing the block length. At the length that would actually have been best on each draw the taper is ahead by 2.12 points of a 35.8% error, at 22.1 paired standard errors; at a length estimated from the sample's own persistence it is ahead by 1.89. At the length this field's own figures use — eight — the rectangle is ahead by 2.04, and at the rule of thumb by 4.54. Every rule sees the same draws. What separates them is the length: the two rules that lose to the rectangle pick 4.00 and 8.00 where the best available is 24.57, and a tapered window at a quarter of the right length has thrown away most of what it was weighting.

An ordering that depends on the rule

The tapered block beats the rectangular one at the best available block length and at one estimated from the data. At a length written into a protocol, and at the rule of thumb, the rectangle wins — at every sample size measured.

feasible · Bootstrap The argument is a third of the size of the thing it is inside. Three quantities on one scale, in points of the error in a block resample's implied long-run variance, at 120 rows. The gap between the two windows at the best available block length — the whole subject of the comparison this field inherited — is 2.12 points. What the best rule a practitioner could actually run gives up against that same best length is 7.26, a factor of 3.42. What the rule of thumb gives up is 26.01. So the ordering between windows is worth establishing and is not worth arguing about, and the sentence that follows from it is not use the taper but estimate the block length, because that is where the points are.

What choosing the length costs

The gap between two block windows at the best available length is 2.12 points. What the best rule a practitioner could run gives up against that same length is 7.26. The argument is a third of the size of the thing it is inside.

feasible · Bootstrap

Named alongside it

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

Bias-varianceBlock bootstrapBlock lengthClosed formDependenceLong-run varianceMean squared errorMonte CarloResamplingTaperingTuning parameterSelection effect

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