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

Jensens inequality — where it appears

That a concave function of an average is at least the average of the function, which is why a random sample size covers less than a fixed one. It is the whole of the shortfall a sequential fixed-width rule has, which is why blinding cannot repair it and makes it worse by making N noisier.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

No block size is best at both things the procedure claims. Two claims and one dial. The honest interval's half-width falls as the blocks get smaller, because the interval's degrees of freedom are the number of blocks: 0.2602 at blocks of two against 0.2933 at blocks of sixteen. The fixed-width claim — that the mean is within 0.25 of the truth — gets more reliable as they get larger, because the sample size is less variable: 92.40% against 95.00%. Both are computed from the same runs, and the second is reproduced to within a tenth of a point by E[2Φ(d√N/σ) − 1], which needs the sample-size distribution and nothing else. The schedules sit at the bottom left: as narrow as the smallest fixed block and as few observations, with the rule's spread estimate on half as many degrees of freedom again.

Two degrees of freedom, one total

The block size is a dial, and the two things a fixed-width procedure claims move in opposite directions along it. Divide the width by the square root of the sample size and one of them turns out to depend on the number of blocks and on nothing else.

pace · Width Three answers to how much sample is left. What a set of inverse-probability weights leaves of the treated arm, by three routes, at six settings of the assignment rule. The integral 1/(π∫φ/e) reads the whole covariate space and falls from 0.9392 to 2.655e-3. Kish's effective size counted in samples of 600 falls only to 0.2861, because almost all of the integral's fall is in a region a sample of six hundred never draws from. And the fraction the variance of the weighted mean actually delivers is lower again — 0.1155 — because the variance is the average of one over the effective size and the effective size averaged is not the same number. At the widest overlap all three agree to 0.05%.

How many observations a weight leaves

Kish's effective sample size is exact — for an outcome whose mean does not move with the covariates the weights are built from, the studentised variance reads 1.0680 where the formula says one. For the population's own outcome the same reading is 6.769, rising to 52.497.

weights · Weighting What the exactness costs, and the dial it is bought with. The median half-width of the interval each rule reports, at a requirement of 0.4 and a first look after 5 observations. The flat line is the interval a practitioner writes at the purely sequential rule's stopping time, which covers 91.72% rather than 95%. The curve is the blinded rule, which covers its nominal level at every block size: it reads b − 1 degrees of freedom where the other reads n − 1, and pays for the exactness in width. The best block size is 3, at 0.4712. Larger blocks give the stopping rule a better estimate and the interval a worse one, and the two costs go opposite ways, which is what puts the minimum in the middle.

What the blindfold costs

The exactly-covering rule pays for it in the width of the interval, and the block size is a dial between two costs that run in opposite directions. And on an interval whose width was fixed in advance, the same repair buys nothing at all.

blind · Nuisance The displacement per parameter, under five predictor laws. The out-of-sample differential one extra parameter costs a correct candidate, at a window of 80 rows, by two routes that share no arithmetic: counted from rolling forecasts over 1,500 draws of the fifteen-candidate table, and implied by the expected leverage of a fresh row over 80,000 draws of the predictors alone. The Gaussian closed form is 0.01386. normal: 0.01382 counted, 0.01386 from the leverage; Student's t, 8: 0.01379 counted, 0.01407 from the leverage; Student's t, 5: 0.01419 counted, 0.01448 from the leverage; Student's t, 3: 0.01751 counted, 0.01780 from the leverage; exponential: 0.01465 counted, 0.01487 from the leverage.

The displacement on a heavy tail

A correct candidate with one more parameter is behind out of sample by a closed-form amount, and the closed form rests on one Gaussian fact: the expected leverage of a fresh row is q/(R − q − 1). On predictors that are not Gaussian it becomes whatever the leverage is, and two routes that share no arithmetic agree on what that is. With a finite fourth moment it barely moves — 1.5% more per parameter at eight degrees of freedom, 4.5% at five, 7.3% on an exponential. At three degrees of freedom it is 28% more, and the estimate of it will not settle: four batches of twenty thousand draws disagree by more than a quarter.

select · Multiplicity The excess over the Gaussian displacement, by window. By how much the expected leverage per parameter exceeds the Gaussian closed form, with the fresh row integrated out, at windows 20, 40, 80, 160, 320, 640. normal: -0.90%, -0.61%, -0.04%, -0.06%, -0.02%, 0.00%. t on eight: 4.14%, 2.78%, 1.79%, 0.73%, 0.28%, 0.18%. t on five: 10.70%, 7.50%, 3.93%, 2.01%, 1.26%, 0.70%. t on three: 44.44%, 32.67%, 22.62%, 16.43%, 11.70%, 7.63%. exponential: 27.91%, 14.01%, 7.21%, 3.53%, 1.93%, 0.85%. On normal predictors the closed form is exact from a window of 80.

An excess and its window

The parameter-count displacement is a Gaussian fact, and on heavy-tailed predictors it runs a few per cent over. Swept across windows from 20 to 640 rows, the excess is about the predictors' excess kurtosis divided by the window: a law with a fourth moment of nine on 80 rows is 7.5% over, and on 640 rows under one. At three degrees of freedom it falls only as the root of the window, from 44% to 7.6%. And the instability once blamed on that law was not there: with the fresh row integrated out, the batches agree to a point and a half, and the mean exists.

select · Multiplicity

Named alongside it

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

Closed formMonte CarloCoverageDegrees of freedomFixed-width intervalHeavy tailKurtosisLeverageModel selectionNuisance parameterSample sizeStopping rule

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