Student's t — where it appears
Named by 14 essays across 7 fields — each of them below, with the objects they name alongside it.
Where the two tails disagree
A 95% t interval on an exponential source at 120 observations covers 94.81%, which reads as very nearly right. It misses below the mean on 4.08% of samples and above on 1.11% — one tail 63% too heavy and the other 56% too light, and the total is the statistic that hides it.
A degrees of freedom that is not a count
The pooled two-sample test's size runs from 0.55% to 18.91% across forty units split five ways against five variance ratios, with a true null in every cell. Welch's runs from 4.63% to 5.51% — bought with a degrees of freedom that is a function of the data, not an integer, and not a count of anything.
The plot is about the wrong quantity
A t interval needs the sampling distribution of the mean to be normal, not the data. A two-lump source leaves its quantile band on 100% of samples of forty and its interval covers 94.80%; a t on three degrees of freedom leaves it on 57% and covers 95.73%, the best of five sources.
The side a bound is read from
On thirty exponential observations the upper limit of a 95% t interval is exceeded by the true mean 6.38% of the time, against the 2.5% a safety margin set from it assumes. Widen the interval until its total coverage is exactly 95% and the upper limit is still exceeded 4.69% of the time. A symmetric repair fixes the number that is reported and not the one that is used; Hall's transformation, which bends the interval, takes the same rate to 3.31%.
A limit from the family it came from
On fifteen exponential observations an upper limit for the mean built from the gamma family's own exact pivot, with the shape estimated by maximum likelihood, is exceeded 4.04% of the time at a mean margin of 0.746 standard deviations. Hall's transformation is exceeded 4.61% at 1.038, and the same family limit with the shape read from the sample skewness is exceeded 17.51%. Knowing the family and estimating its one parameter well recovers most of what knowing the population buys — and the parameter is estimated well only from the logarithms.
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.
The correction for not knowing the spread
The t distribution exists because the standard deviation is estimated rather than known. At eight observations, using the normal instead makes every interval 12% too short — and the coverage that follows can be measured rather than argued about.
The level a limit should be set at
When a failure costs twenty times a standard deviation of margin, a t limit on fifteen exponential observations set at the conventional 97.5% has an expected loss of 2.143. Set at 99.95%, the same formula's loss is 1.263 — within 1% of the best any fixed multiple of s/√n can do, and ahead of both Hall's transformation and the fitted gamma family at their own best levels. Hall's is the worst at every level on every source. Choosing the level does more than choosing the construction, and the correction that reads the sample is the one no level rescues.
The correction a t test would use
On thirty exponential draws the one skewness term that held a sum's critical value within 10% out to five standard deviations nearly triples a t test's 1% size when the source's skewness is known, and its upper critical value turns back — at a level of 0.89% on ten draws, where a test asked for one in a thousand rejects 121 times too often. Hall's monotone cubic removes the turn and gains a floor. Built from the sample's own skewness it holds the short side within 20% from 10% down to one in a hundred thousand, and leaves the long side, the one a safety limit reads, at twice its level: the samples that fail are the ones that hide their skewness.
A failure charged by its size
Charge an upper limit's failure by how far the true mean exceeds it rather than by a fixed amount, and the level a limit should be set at falls further than any other change in this comparison moves it: on fifteen exponential observations, a penalty of twenty per standard deviation of shortfall asks for a t limit at 97% where a penalty of twenty per failure asked for 99.95%. The failures are small whichever limit produces them — about an eighth of a standard deviation — so a charge by size is a small charge per failure. The ranking hardly moves: the t limit stays within about 1% of the fitted family, whose failures are no smaller than its own, and Hall's transformation stays last by 11%.
A tilt for the ratio
A t statistic is a function of two means, so its saddlepoint tilts the joint law of each draw and its square. On an exponential source every such tilt is a truncated normal, and none has a coefficient of variation above one — the exponential is the edge of its own family. Integrated over the samples a tilt can reach, the saddlepoint reads a t statistic's tails on thirty draws within 2.5% from 10% to one in a million on the long side, with nothing fitted. The samples it cannot reach are a third of all samples, they sit at the centre of the statistic rather than in its tails, and a larger sample has more of them.
Weights from an approximate law
A companion whose law is only approximately known can still be cut into bands, with each band's weight computed from the approximation. The weight's error is a bias that does not shrink, and the run multiplies it: the stratified rate is worth p(1 − p)/(v + Nb²) times the draws. Reading Student's t on nineteen degrees of freedom as normal breaks even at 669 draws; the normal with the t's variance at about fifteen thousand; Fisher's one-term expansion at twenty-four million. What decides the bias is how the weight errors line up with the band rates, not how large they are — Fisher's is wrong by sixty per cent in one band and biases the rate by six hundredths of one.
A level set before the source is known
Every best level for an upper limit on skewed data was found knowing the source, and the fifteen observations that would tell the sources apart are the ones being used to set the limit. Chosen instead to protect the worst of four skewed sources, the t limit's level is 97.25% when a failure is charged by the size of its shortfall, and it costs at most 0.8% more than each source's own best — the sources disagree about the best level and agree that missing it a little is cheap. The fitted gamma family, the construction meant to adapt to the source, is the one that needs it: its best levels run from 90% to 96.5% and its minimax level costs 3.4%. Charged a fixed amount per failure, the t limit's minimax level is 99.97% and costs 6.5%; the conventional 97.5% costs up to 119.9%.
A prior on the spread between two sites
Integrate the between-site spread of a two-site study under a half-normal prior of scale 1 and the interval for the overall mean is 3.01 wide and covers 94.63% when the true spread is 1 — the supplied interval again, under another name. At a true spread of 3 it covers 68.38%. Every prior tried covers 95% averaged over its own draws and none covers it at every spread; each holds up to about its own scale, because two sites move the posterior median of the spread by less than a third of what the truth moves.
Named alongside it
The objects these essays reach for when they reach for this one.
SkewnessCoverageSample sizeUpper confidence limitEstimated varianceDecision theoryDegrees of freedomEdgeworth expansionExpected lossMonte CarloNormalitySignificance level