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The Gumbel distribution is commonly used in extreme value theory, which is the study of the behavior of extreme events or outliers. Introduced by Emil Julius Gumbel in 1954 [1], it is also known as the Type I extreme value distribution.
The Gumbel distribution is used to model the distribution of the maximum or minimum value from a sample of independent and identically distributed random variables. It is well-suited for modeling extreme events because it has long tails, meaning that it assigns a higher probability to rare events than other symmetric distributions.
The Gumbel distribution has a number of applications in fields such as environmental science, finance, and engineering. For example, it can be used to model the distribution of extreme temperatures, river flows, wind speeds, and other meteorological variables.
Gumbel distribution properties
The probability density function (PDF) of the Gumbel distribution is given by:
where
- z = (x – μ) / β
- μ = location parameter
- β = scale parameter.
The cumulative distribution function (CDF) is F(x) = e-e (x – μ) / β.
The mean is μ + βγ, where γ is Euler’s constant.
History of the Gumbel distribution
The Gumbel distribution is named after Emil Julius Gumbel, a German mathematician and statistician who lived from 1891 to 1966. Gumbel created the distribution in the 1950s while studying extreme value theory. Specifically, he was interested in finding a distribution that could approximate the distribution of the maximum or minimum value from a large sample of independent and identically distributed random variables.
Gumbel’s work on the Gumbel distribution became popular in part because it provided a way to model and understand extreme events, which are often of particular interest in fields such as engineering, finance, and environmental science. In the years following Gumbel’s initial research, many applications of the distribution were discovered, and it is now widely used in a range of fields.
One specific example of an extreme event that is often modeled using the Gumbel distribution is the occurrence of high annual maximum rainfall. This is important for designing infrastructure such as stormwater management systems and dams, as well as predicting flood risks in a given area.
As an example of the Gumbel distribution being used to model extreme events, the paper “Modeling extremes using the generalized extreme value distribution” by Coles [2] provides a comprehensive overview of extreme value theory and includes numerous examples of the Gumbel distribution being used to model a variety of extreme events for applications such as flood risk management, wind damage prediction, and earthquake analysis.
Gumbel’s work was one of the early breakthroughs in the field of extreme value theory, which has since become an important area of research in mathematical statistics.
References
[1] Gumbel, E. J. (1954). “Statistical theory of extreme values and some practical applications.” Applied Mathematics Series (No. 33). US Government Printing Office.
[2] Coles, S. G. (2001). Modeling extremes using the generalized extreme value distribution. Extremes, 3(3), 287-301.