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

Expected shortfall — where it appears

The average amount by which a quantity falls past a limit when it does, rather than how often it does. Charging a failure by its shortfall rather than by its occurrence makes small misses cheap, which moves the level a limit should be set at and shrinks the cost of choosing it wrongly.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Expected loss of three upper limits against the level each is set at, when a failure is charged by its size, fifteen exponential observations, penalty 20. A failure costs 20 times the distance, in standard deviations, by which the mean exceeds the limit. At the conventional 97.5% the t limit's loss is 0.728, Hall's 1.136 and the fitted family's 0.835. Each at its own best level: the t limit 0.726 at 97.00%, Hall's 0.805 at 90.00%, the family 0.735 at 92.50%.

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%.

tails · Student What it costs to set the t limit at one level without knowing which skewed source the 15 observations came from. For the t limit on 15 observations, charged twenty per standard deviation of shortfall plus its margin, the loss at each nominal level as an excess over the best level for each source. The best levels are 97% for the exponential, 96.25% for the gamma, shape 2, 97.75% for the gamma, shape ½, 98.25% for the lognormal. The minimax level, 97.25%, costs at most 0.8% on any of them: 0.1% on the exponential, 0.6% on the gamma, shape 2, 0.2% on the gamma, shape ½, 0.8% on the lognormal. The conventional 97.5% costs up to 1.0%.

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%.

tails · Student

Named alongside it

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

Decision theoryExpected lossSample sizeSignificance levelSkewnessStudent's tUpper confidence limit

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