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

Quantile — where it appears

The value below which a stated share of a distribution falls; a test's critical value is a quantile of its null distribution. Reading a statistic at a quantile rather than at its mean is what a test does, and two statistics can agree on average and disagree there.

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

An overlap that depends on where it is read. Each pair's overlap on the doubled log-likelihood scale — the sum of the two searches' own quantiles less the joint search's quantile, as a share of the smaller — at five quantile levels, over 1200 draws under the first-order autoregression. two dictionaries of independent columns: -0.176 at 0.5, -0.000 at 0.75, 0.238 at 0.9, 0.320 at 0.95, 0.538 at 0.99; a break, and an independent column: -1.211 at 0.5, -0.406 at 0.75, -0.029 at 0.9, 0.205 at 0.95, 0.161 at 0.99; two dictionaries of step columns: -0.397 at 0.5, 0.145 at 0.75, 0.438 at 0.9, 0.541 at 0.95, 0.682 at 0.99; a break, and a step column it contains: 1.000 at 0.5, 1.000 at 0.75, 1.000 at 0.9, 1.000 at 0.95, 1.000 at 0.99; a break, and a whitening window: 0.512 at 0.5, 0.640 at 0.75, 0.716 at 0.9, 0.732 at 0.95, 0.754 at 0.99. Every pair but the containment overlaps more at the 95% point than at the median, the control included.

The overlap at the critical value

Two searches over independent noise columns are additive on average: the joint search finds what the two find separately, to within a fortieth of the smaller. At the point a test actually uses — the joint statistic's 95% quantile — the same pair overlaps by 0.320, and a rule that adds the two searches' own critical values rejects 1.4% of true nulls instead of 5%. Nothing about the searches causes it. Quantiles of a sum do not add, and on every pair in the field the overlap at the critical value is larger than the overlap the field measured.

apart · Criterion The overlap climbs and the interaction does not. The over-charge at each quantile level split into overlap — the two searches' own quantiles less the sequential run's — and interaction — the joint search's quantile less the sequential run's — for three pairs over 1200 draws. two independent dictionaries: overlap -0.40, 0.01, 1.55, 2.44, 5.72; interaction 0.07, 0.01, 0.11, 0.06, 0.00. two step dictionaries: overlap -0.73, 8.19, 22.53, 31.02, 47.75; interaction 4.87, 4.55, 4.90, 4.75, 1.10. a break and a window: overlap 17.88, 29.52, 43.01, 53.79, 71.61; interaction 1.57, 1.29, 1.19, 3.22, 2.75, at the 50%, 75%, 90%, 95% and 99% points.

The split at the critical value

Split the over-charge of two searches at the joint statistic's 95% point rather than at its mean, and the two components behave differently. The interaction — what only the joint search can reach — stays within two units of its average at every level up to the 95% point. The overlap climbs from nothing to most of the over-charge, on two searches over independent columns as much as on searches that share ground. At the critical value the split's overlap has stopped meaning shared ground: it is mostly the arithmetic of adding tails, and the identity that defines it hides that.

separate · Break point A threshold read along the skew. What a rule balancing a threshold at the value 1 on each covariate's own scale removes of the interaction between the thresholds, against the covariate's skewness, on five copulas at a Spearman rank correlation of 0.4. Gaussian: 16.41% with no skew, 10.07% at a skewness of 11.16; t on four: 22.18% with no skew, 12.11% at a skewness of 11.16; Frank: 13.19% with no skew, 10.14% at a skewness of 11.16; Clayton: 5.33% with no skew, 4.82% at a skewness of 11.16; Clayton turned over: 33.34% with no skew, 17.61% at a skewness of 11.16. Every leak falls, because the value moves towards the median.

Where the skew puts a threshold

The field read the mean-balancing rule along a sequence of skewness and left the threshold rule — a cut at a fixed value on the covariate's own scale — unread. Read, it turns out to be no new curve. Skewing the scale moves the value 1 from the 84th percentile of the latent covariate to the 74th, and the rule's leak is the copula field's cut sweep read at that quantile: it falls on every copula, from 16.41% to 10.07% under a Gaussian. Whether skew and dependence compound or cancel is then a fact about the slope of each copula's curve, and unlike the mean rule's, no cell changes its answer along the sequence.

stronger · Adjustment 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 One order's interaction climbs under long memory. The break-and-window pair's interaction — the joint search's quantile less the sequential run's — at five quantile levels, in the field's order (break first, solid) and the other (window first, dashed), under three laws, 1200 draws each. first-order autoregression: break first 1.57, 1.29, 1.19, 3.22, 2.75; window first 0.98, 1.26, 1.53, 2.55, 2.14. five-period moving average: break first 2.00, 1.88, 2.29, 2.63, 0.60; window first 0.16, 0.21, 0.26, 0.29, 0.00. long memory: break first 0.53, 0.80, 1.05, 0.46, 1.35; window first 2.01, 1.62, 4.75, 6.07, 7.85, at the 50%, 75%, 90%, 95% and 99% points. Under two of the laws, and in the field's order under all three, the interaction stays within a few units of its median. Under long memory with the window searched first it climbs from 2.01 to 7.85, against an average of 1.99; the same order's averages under the other two laws are 0.94 and 0.17.

A tail the law gives the pair

Split the over-charge of two searches at their critical value and the interaction — the configurations only the joint search reaches — stays near its average while the overlap climbs. That held under one law. Under a five-period moving average and under long memory it holds for every pair in the order the field reads, to within three units. In the other order, one pair acquires a tail of its own: with the window searched first and the break second, the interaction under long memory climbs from 2.0 at the median to 7.9 at the 99th percentile, four times its average.

separate · Break point

Named alongside it

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

Monte CarloChange pointCombinatorial searchCritical valueData snoopingLikelihood ratioConvolutionSkewnessSpecification searchSupremum statisticAutoregressionCalibration

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