Exceedance Distribution


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The exceedance distribution tells us how often we can expect rare events.

The exceedance distribution function (EDF) is defined as 1 minus the cumulative distribution function (CDF) [1]:

EDF = 1 – CDF

The EDF, like the CDF, is bounded between 0 and 1. The CDF tells us what fraction of events are below a specified value. For example, if a measure is at P = 100m with a CDF of 0.4, that tells us 40% of the values are below 100m; the EDF is 1 – 0.4 = 0.6, so 60% of values are above 100m. Because of its complementary nature, the EDF is sometimes called the complementary distribution function.

Heavy tails in an exceedance distribution tell us that there are more occurrences of extreme events; these values represent a lower quantile than a light-tailed distribution [2].

Sometimes the exceedance curve – the graph of a continuous probability exceedance distribution – is called a risk curve or expectation curve.

Exceedance Distribution Formula

Various formulas have been proposed in the literature.

For example, Sarkadi [3] provided the following formula for the “number of exceedances”,

Where

  • n and N are independent observations (two sample sizes)
  • ξ is the number of exceedances, or the number of elements of sample size N which surpass (are larger than) at least nm  + 1 elements of the sample size n( 1 ≤mn).

Several other authors reference the same formula (e.g., [4]). The formulas have largely become somewhat obsolete with the availability of computer software.

References

[1] Katul, G. Intensity-Duration-Frequency Analysis. Online: https://nicholas.duke.edu/people/faculty/katul/ENV234_Lecture_7.pdf

[2] do Nascimento & Pereira (2017). A Bayesian approach to extended models for exceedance. Brazilians Journal of Probability and Statistics. Vol 31, No. 4, 801-820.

[3] Sarkadi, K. (1957). On the Distribution of the Number of Exceedences. The Annals of Mathematical Statistics. Vol 4. P.1021.

[4] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[5] Lundberg, O. (1940). On random processes and their applications to sickness and accident statistics. Almqvist, Uppsala.