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

Heavy tail — where it appears

A distribution whose extreme values are common enough to dominate its averages. Where the tail is heavy enough the mean may not exist at all, and then a sample average does not settle as the draws increase while a sample median does.

Named by 13 essays across 7 fields — each of them below, with the objects they name alongside it.

A weak instrument gives back the problem it was hired for. The counted mean bias of two-stage least squares at 4 instruments and 200 rows, over 2000 draws a setting, against the standard approximation and against the least-squares inconsistency the instrument was brought in to remove. At π = 0.02 the counted bias is 0.3220 ± 0.0142 where least squares is out by 0.3594 — 89.6% of the way back. At π = 0.3 it is 0.0118 against 0.2647. The approximation, the inconsistency over the population first-stage F, tracks the count at the weak end and sits above it in the middle: 0.968, 0.971, 0.918, 0.810, 0.740, 0.722, 0.846 as the ratio of counted to approximated bias.

Weak, and back where it started

A consistent instrumental estimate at two hundred rows and a concentration parameter of 0.32 is biased by 0.3220 ± 0.0142 against a least-squares inconsistency of 0.3594 — 89.6% of the way back to the problem it was hired to solve. Just identified, it has no mean at all, and that is measured as a rate rather than assumed.

instrument · Exclusion The estimator has no upper bound on what it costs. What a thinning overlap does to a stabilised inverse-probability estimate of an average effect of 1.0000, over 600 samples of 600 at each of six settings. The spread rises from 0.1965 to 0.8122 and the root mean square error from 0.1964 to 0.9219, so at the thin end the error is very nearly the whole of the quantity being estimated. The lower line is the share of the arm's weighted total the single largest observation owns, averaged over the same draws: 0.69% to 9.78%, and in the worst single draw of the sweep 82.75%. Coverage of the 95% interval goes from 93.7% to 55.5%.

The region with no comparison

A trimmed interval covers the average effect over everybody 90.8% of the time at six hundred rows and 41.0% at nine thousand six hundred, while covering the average effect over the units it kept 94.3% and 96.0% throughout. An interval that gets worse as the sample grows is an interval about something else.

weights · Weighting Either model is enough; neither is not. The bias of three estimators of an average effect of 1.0000, over 600 samples of 600 units with the assignment rule at strength 1, in each of the four cells made by getting each nuisance model right or wrong. The wrong model in both cases is one that omits the second covariate, which the outcome and the assignment both depend on. The outcome model alone is off by 0.8064 whenever it is the wrong one; weighting alone is off by 0.8190 whenever the propensity model is. The augmented estimator built from both is off by -0.0085, -0.0083 and -0.0016 in the three cells where at least one of them is right, and by 0.8118 in the fourth — which is between its two components rather than better than either.

Either model, but not neither

The augmented estimator's bias is −0.0085, −0.0083 and −0.0016 wherever one nuisance model is right, against components off by 0.8064 and 0.8190. One step past the overlap sweep it is the least biased estimator on the table at 0.0857 and the worst on it at 1.9265.

weights · Weighting 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 Which side the misses fall on, skewed shocks, 95.0% intervals. With skewed shocks and the model's parameters known, the share of 10,000 forecasts that fall below and above a nominal 95.0% normal interval, at each horizon: 1 ahead, 0.0% below and 5.2% above; 2 ahead, 0.0% below and 4.9% above; 4 ahead, 0.0% below and 4.6% above; 8 ahead, 0.0% below and 4.2% above; 12 ahead, 0.0% below and 4.9% above.

Shocks that are not normal

Every forecast interval in this field reads a normal quantile, and every series it was scored on had normal shocks. Give a first-order autoregression shocks with heavy tails and the 95% interval still covers about 95%; its 99% interval covers 98.2%. Give it skewed shocks and the 95% interval covers 94.8% one step ahead with every miss on one side — 0.0% below, 5.2% above — and the horizon does not repair it: twelve steps ahead the misses are still 0.0% below and 4.9% above, because a persistence of 0.7 keeps the error's skewness at 1.11 however far out the forecast goes. A two-sided coverage count cannot see any of this.

forecast · Forecast The top of a heavy-tailed population keeps its lead; the top of a light-tailed one gives it back. Select the top share on the first reading and read the group again: the share of its mean lead the second reading keeps, by integration over the true score (lines) and counted on 400,000 draws a parent in 20 batches (points, with two standard errors). The normal keeps exactly 0.6 at every selection. At the top half the Laplace keeps 0.541, the t 0.535 and the uniform 0.648 — the heavy tails keep LESS than the correlation. By the top one per cent the order has reversed: 0.761, 0.784 and 0.443. At one in ten thousand the t keeps 0.977 and the uniform 0.346.

A lead that a heavy tail keeps

Four populations whose readings all correlate at exactly 0.6, and whose least-squares slopes all read 0.6. Select the top one per cent on one reading and measure them again: they keep 60% of their lead if the true scores are normal, 76.1% if they are Laplace, 78.4% if they are a t on four degrees of freedom — and 44.3% if they are uniform. The correlation predicts the regression of the extremes for one shape of population only.

paradox · Rtm Estimating P(Z > 5) = 2.8665×10⁻⁷ with plain draws and with four proposals. One seed each. Plain simulation draws nothing past 5 in 100,000 and estimates zero throughout. At 100,000 draws the proposal N(5, 1) reads 1.009 of the truth, N(9, 1) 1.041, N(4.5, 0.25²) 1.039 and N(5, 0.3²) 1.008. Values above 2.2 are drawn at the top edge.

The draws aimed at the tail

The chance a standard normal exceeds 5 is 2.8665×10⁻⁷, and a plain simulation needs 349 million draws to estimate it to within ten per cent. Draws aimed at the tail and weighted back need 565. Aimed slightly too narrowly, the same method has an infinite variance, an interval that covers 86.0% and gets worse with more draws, and an effective sample size that reads healthier than a proposal that works.

method · Seeds Three intervals as one strength is spread thinner, at a concentration of 8. Coverage of three nominal 95.0% intervals on the same 1000 draws of 200 rows at each count, when a total concentration parameter of 8 is spread over 1 to 32 instruments. Two-stage least squares covers 97.2%, 96.4%, 94.0%, 86.7%, 73.2%, 51.5%. Building each row's fitted treatment from a first stage that never saw that row covers 97.1%, 97.2%, 98.3%, 97.8%, 97.9%, 98.7%. Limited-information maximum likelihood with its conventional standard error covers 97.2%, 96.7%, 95.7%, 90.9%, 85.2%, 79.0%. At one instrument the likelihood estimator is two-stage least squares exactly, which is why the first readings of those two agree to the last draw.

Leaving each row out of its own first stage

Spread a fixed first-stage strength over thirty-two instruments and two-stage least squares covers 51.5%. Build each row's fitted treatment from a first stage that never saw that row and the same draws cover 98.7% — through an interval 5.99 times as wide, around an estimate that misses by more than the whole effect on 34.7% of draws. At eight times the strength the same repair covers 95.3% and costs a width factor of 1.66.

instrument · Exclusion Which side the misses fall on. Misses of a nominal 95% forecast interval 1 step ahead on 60 observations of a first-order autoregression at 0.7, split by side. normal shocks: the interval as printed 3.1% below and 3.1% above, shocks from the residuals 3.8% below and 4.2% above, the same, coefficient re-estimated 3.3% below and 4.4% above. Student's t on five: the interval as printed 3.3% below and 3.0% above, shocks from the residuals 3.9% below and 3.0% above, the same, coefficient re-estimated 3.6% below and 3.1% above. skewed shocks: the interval as printed 0.0% below and 6.5% above, shocks from the residuals 4.3% below and 4.1% above, the same, coefficient re-estimated 1.8% below and 4.3% above.

A tail sixty residuals have barely seen

A forecast interval that reads a normal quantile has every one of its misses on one side when the shocks are skewed, and the natural repair is to draw future shocks from the model's own residuals. It works on the side: 0.0% below and 6.5% above becomes 4.3% and 4.1%. It pays for that with coverage, 93.5% to 91.6%, and on sixty observations it does this under normal shocks too. And a nominal 99% interval has a ceiling nothing in the resampling can lift, because one new shock beats all of sixty old ones in either direction with probability 3.3%.

forecast · Forecast 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 Twenty studies of a thousand readings estimate the regression of the extremes: the t, four degrees parent. Each thin line is one study of 1000 readings from the t, four degrees parent: Tweedie's formula with the log-density's slope estimated by a degree-5 log-spline, drawn up to that study's largest reading. The thick line is the share kept by integration over the true score, and the dashed line the correlation, 0.6. The top ten readings of the median study begin at 2.45. Over 400 studies the corrected share kept by the top one per cent averages 0.8190, with a spread of 0.0806, against 0.7842 by integration; a Gaussian kernel averages 0.7914 with a spread of 0.0853.

The slope of a density nobody can see

Tweedie's formula corrects a reading by the slope of the readings' own log-density, and a study has its readings. Estimated from a thousand of them, the correction for the top one per cent beats the correlation's linear rule on 84.0% to 98.0% of studies from heavy-tailed populations and on 75.0% to 81.5% from a bounded one — and costs an error of 0.09 to 0.12 where the population is normal and the rule was already exact. At 250 readings the log-spline loses to the rule it replaces, and at 16,000 the same log-spline gets worse on a power tail.

paradox · Rtm How many people a study has to retest before its estimated correction is worth having, normal errors. Root mean squared error, over 400 studies of a thousand readings at each retest size, of the share of the top one per cent's lead a second reading keeps: Tweedie's correction with a kernel log-slope and the error variance estimated from the retested people (solid), and the linear rule at the reliability those people estimate (dashed). Laplace parent: 0.121, 0.108, 0.084, 0.086, 0.084, 0.078 against 0.343, 0.259, 0.201, 0.183, 0.171, 0.164. T, four degrees parent: 0.133, 0.103, 0.086, 0.084, 0.080, 0.078 against 0.358, 0.278, 0.235, 0.212, 0.205, 0.187. Uniform parent: 0.247, 0.192, 0.156, 0.140, 0.118, 0.117 against 0.260, 0.208, 0.185, 0.166, 0.160, 0.157 — at 10, 20, 50, 100, 200, 1000 retested.

A correction built on a retest

Tweedie's correction for regression to the mean needs the error variance, and a study gets it from a few people measured twice. Twenty of them are enough: with normal errors the estimated correction then beats the correlation's rule on 73% to 77% of studies of a thousand readings from a heavy-tailed population, though its variance is uncertain by a third. What twenty cannot reveal is the error's shape. Laplace errors of the same variance make the correction overstate the lead a top group keeps by about 0.17, the plain rule then wins on most studies, and a study needs about two hundred retest pairs before a check on its own differences notices.

paradox · Rtm A prior centred on the truth's population and too confident about it: coverage across the population, and how often the check flags it. Truths drawn from a population of proportions centred at 0.2 and worth ten observations; twenty trials per study; the analyst's prior centred at 0.2 with the weight on the axis. Coverage of the 95% credible interval, exact over the population: 94.7% at 2, 96.0% at 5, 95.0% at 10, 88.1% at 20, 75.9% at 35, 61.3% at 60, 48.2% at 100, 34.0% at 200, 23.9% at 400. Share of studies whose count the prior-predictive check puts in its 5% tail: 0.1%, 1.8%, 3.2%, 14.2%, 17.3%, 17.3%, 21.9%, 21.9%, 21.9%.

A prior wrong about its own spread

A prior can be centred exactly where the truths lie and still be wrong about how far they spread. Ten times too confident — worth a hundred observations about proportions whose population is worth ten — its 95% credible intervals cover 48.2% of the population's studies at twenty trials, and nothing in the interval shows it: it is 0.456 of a flat prior's width, against 0.436 for a prior whose hundred observations' confidence is earned. A prior with tails too thin misses the one study in ten that comes from somewhere unusual a third of the time, and a flat component of two per cent brings that study's coverage from 65.6% to 84.0%.

bayes · Credible

Named alongside it

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

Monte CarloClosed formCoverageConfidence intervalForecast errorKurtosisLeverageMeasurement errorRegression to the meanTweedie's formulaAutoregressionAverage treatment effect

All concepts