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# Gumbel Distribution

The Gumbel distribution is used to model the largest value from a relatively large set of independent elements from distributions whose tails decay relatively fast, such as a normal or exponential distribution. As a result, it can be used to analyze annual maximum daily rainfall volumes. In this way, it can be used to predict extreme events such as floods, earthquakes or hurricanes.

For this reason, the Gumbel distribution is also called the extreme value type I distribution and is used to find a maximum extreme value. Setting x to –x will find the minimum extreme value.

The pdf of the Gumbel distribution with location parameter μ and scale parameter β is

where β > 0. The cdf is

The inverse of the Gumbel distribution is

The standard Gumbel distribution is the case where μ = 0 and β = 1.

The Gumbel distribution is sometimes called the double exponential distribution, although this term is often used for the Laplace distribution.

If x has a Weibull distribution, then -ln(x) has a Gumbel distribution.

Key statistical properties of the Gumbel distribution are:

Figure 1 – Statistical properties of the Gumbel distribution

Here, γ is the Euler-Mascheroni constant whose value is –ψ0(1), the negative of the digamma function at 1 (see MLE Fitting Gamma Distribution) with a value approximately equal to .577215665.

Figure 2 shows a graph of the Gumbel distribution for different values of μ and β.

Figure 2 – Chart of the Gumbel distribution

Real Statistics Functions: The Real Statistics Resource Pack provides the following functions for the Gumbel distribution.

GUMBEL_DIST(x, μ, β, cum) = the pdf of the Gumbel distribution f(x) when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.

GUMBEL_INV(p, μ, β) = the inverse of the Gumbel distribution at p

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