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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*n*–*m*+ 1 elements of the sample size*n*( 1 ≤*m*≤*n*).

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.