Generalized extreme value distribution
The generalized extreme value distribution (GEV) is a three-parameter family of continuous probability distributions that models the asymptotic distribution of block maxima or minima from large samples of independent and identically distributed random variables, unifying the Gumbel, Fréchet, and reversed Weibull distributions into a single form.[1] Its cumulative distribution function is given by G(z; \mu, \sigma, \xi) = \exp\left\{ -\left[1 + \xi \frac{z - \mu}{\sigma}\right]^{-1/\xi}_+ \right\} for \xi \neq 0, where \mu \in \mathbb{R} is the location parameter, \sigma > 0 is the scale parameter, \xi \in \mathbb{R} is the shape parameter, and the subscript + denotes the positive part function (i.e., \max(0, \cdot)); for \xi = 0, it reduces to the Gumbel form G(z; \mu, \sigma, 0) = \exp\left\{ -\exp\left( -\frac{z - \mu}{\sigma} \right) \right\}.[1] The support is z \in \{ w : 1 + \xi (w - \mu)/\sigma > 0 \}, ensuring the argument of the outer exponent is well-defined.[1] This distribution emerges as the limiting law under the Fisher–Tippett–Gnedenko theorem (also known as the extremal types theorem), which characterizes the possible non-degenerate limits of the normalized sample maximum (or minimum) from distributions in the domain of attraction of an extreme value limit; specifically, the theorem proves that such limits must belong to one of three types—Gumbel (\xi = 0), Fréchet (\xi > 0), or reversed Weibull (\xi < 0)—all encompassed by the GEV family.[2] The theorem, originally developed through contributions from Fisher and Tippett (1928) and rigorously proven by Gnedenko (1943), provides the theoretical foundation for extreme value theory, analogous to the central limit theorem but for extremes rather than averages.[2] The GEV was first unified and parameterized in this form by Jenkinson in 1955, initially for analyzing annual maxima of meteorological variables such as rainfall and wind speeds.[3] Key properties of the GEV include its flexibility in capturing tail behaviors: positive \xi yields heavy-tailed distributions suitable for unbounded extremes with power-law decay, negative \xi produces finite upper (or lower) endpoints for bounded supports, and \xi = 0 describes exponentially decaying tails.[1] Parameter estimation typically involves methods like maximum likelihood or probability-weighted moments, though challenges arise with small samples or near-boundary \xi values due to the distribution's asymmetry and potential for heavy tails.[4] In practice, the GEV is applied across fields to quantify risks from rare events, including flood frequency analysis in hydrology, storm surge modeling in meteorology, extreme loss prediction in finance, and seismic event assessment in geophysics, enabling probabilistic forecasts and infrastructure design under uncertainty.[5]Definition and Specification
Cumulative distribution function
The cumulative distribution function (CDF) of the generalized extreme value (GEV) distribution provides a unified mathematical framework for modeling the distribution of block maxima (or minima) in extreme value theory. It is defined for a random variable Z as G(z; \mu, \sigma, \xi) = \exp\left\{ -\left[1 + \xi \frac{z - \mu}{\sigma}\right]^{-1/\xi} \right\} for \xi \neq 0, where the support condition requires $1 + \xi (z - \mu)/\sigma > 0.[3] This form was first proposed by Jenkinson to combine the three classical extreme value distributions into a single family.[3] When the shape parameter \xi = 0, the CDF takes the limiting form G(z; \mu, \sigma, 0) = \exp\left\{ -\exp\left[ -\frac{z - \mu}{\sigma} \right] \right\}, which corresponds to the Gumbel distribution and is defined over the entire real line (-\infty < z < \infty).[5] The parameters in the GEV CDF play distinct roles: the location parameter \mu \in \mathbb{R} shifts the distribution horizontally, the scale parameter \sigma > 0 controls the spread, and the shape parameter \xi \in \mathbb{R} governs the tail heaviness.[5] The domain of support varies with \xi. For \xi > 0, the distribution has a lower bound at z > \mu - \sigma / \xi and extends to +\infty, reflecting heavy-tailed behavior suitable for maxima from distributions with unbounded support. For \xi < 0, the support is upper-bounded at z \leq \mu - \sigma / \xi with extension to -\infty, appropriate for modeling bounded upper extremes.[5]Probability density function
The probability density function (PDF) of the generalized extreme value (GEV) distribution, denoted g(z; \mu, \sigma, \xi), is obtained by differentiating the cumulative distribution function (CDF) with respect to z.[6] For \xi \neq 0, g(z; \mu, \sigma, \xi) = \frac{1}{\sigma} \left[ 1 + \xi \frac{(z - \mu)}{\sigma} \right]^{-\frac{1}{\xi} - 1} \exp\left\{ -\left[ 1 + \xi \frac{(z - \mu)}{\sigma} \right]^{-\frac{1}{\xi}} \right\}, defined where $1 + \xi (z - \mu)/\sigma > 0.[7] In the limiting case \xi = 0, g(z; \mu, \sigma, 0) = \frac{1}{\sigma} \exp\left[ -\frac{(z - \mu)}{\sigma} \right] \exp\left\{ -\exp\left[ -\frac{(z - \mu)}{\sigma} \right] \right\}, defined for all real z.[7] To derive the PDF, consider the CDF G(z; \mu, \sigma, \xi) = \exp\left\{ -\left[ 1 + \xi (z - \mu)/\sigma \right]^{-1/\xi} \right\} for \xi \neq 0. Let u = 1 + \xi \frac{z - \mu}{\sigma}. Then G(z) = \exp\left( -u^{-1/\xi} \right), so \log G(z) = -u^{-1/\xi}. Differentiating, \frac{d}{dz} \log G(z) = \frac{g(z)}{G(z)} = \frac{1}{\xi} u^{-1/\xi - 1} \cdot \frac{\xi}{\sigma} = \frac{1}{\sigma} u^{-1/\xi - 1}. Thus, g(z) = G(z) \cdot \frac{1}{\sigma} \left[ 1 + \xi \frac{z - \mu}{\sigma} \right]^{-1/\xi - 1}. The case \xi = 0 follows by taking the limit of the general PDF or directly differentiating the Gumbel CDF.[6] The asymptotic tail behavior of the PDF varies with \xi. For \xi > 0, the right tail is heavy, asymptotically following a power-law decay similar to the Fréchet distribution, with unbounded support to the right.[8] For \xi < 0, the support is bounded above at \mu - \sigma / \xi, leading to a finite endpoint and rapid decay near the boundary, akin to the reversed Weibull case.[9] For \xi = 0, the tails are light, with exponential decay on the right, characteristic of the Gumbel distribution.[6]Parameter interpretations
The generalized extreme value (GEV) distribution features three parameters that provide a unified framework for modeling block maxima or minima in extreme value theory: the location parameter μ, the scale parameter σ, and the shape parameter ξ. This parametrization was introduced by von Mises in 1936, who established sufficient conditions for convergence to limiting extreme value distributions, and independently derived in a unified form by Jenkinson in 1955 to encompass the Gumbel, Fréchet, and Weibull cases.[5][3] The location parameter μ, ranging from -∞ to ∞, represents the central tendency of the distribution and shifts it horizontally. When ξ = 0, μ specifically corresponds to the mode, serving as a reference point for the characteristic largest value in the sample.[1][10] The scale parameter σ, constrained to be positive (σ > 0), quantifies the dispersion or spread of the distribution around μ. It controls the rate of tail decay, with larger values indicating greater variability in extreme observations.[1][10] The shape parameter ξ, ranging from -∞ to ∞, determines the overall form and tail characteristics of the distribution. For ξ > 0, the GEV exhibits Fréchet-like heavy tails, unbounded above, suitable for modeling unbounded extremes with power-law decay. When ξ < 0, it follows a Weibull-like form with a finite upper endpoint at μ - σ/ξ, reflecting bounded maxima. The case ξ = 0 yields Gumbel-like exponential tails, providing a light-tailed alternative.[1][10] Qualitatively, ξ influences the skewness and kurtosis of the GEV distribution through its effect on tail heaviness. Positive ξ enhances positive skewness by emphasizing the right tail, while negative ξ induces negative skewness due to the upper bound. As |ξ| increases from zero, kurtosis generally rises, reflecting heavier tails and more pronounced extreme events.[11]Connections to Classical Extreme Value Distributions
Gumbel distribution case
The generalized extreme value (GEV) distribution with shape parameter ξ approaching 0 reduces to the Gumbel distribution, a classical extreme value distribution that arises as the limiting distribution for maxima of sequences from light-tailed parent distributions.[12] In this limiting case, the parameters μ and σ of the GEV retain their interpretations as the location and scale parameters of the Gumbel distribution, respectively.[12] The cumulative distribution function (CDF) of the Gumbel distribution is given by F(x; \mu, \sigma) = \exp\left( -\exp\left( -\frac{x - \mu}{\sigma} \right) \right), for x \in \mathbb{R} and σ > 0.[13] The corresponding probability density function (PDF) is f(x; \mu, \sigma) = \frac{1}{\sigma} \exp\left( -\frac{x - \mu}{\sigma} - \exp\left( -\frac{x - \mu}{\sigma} \right) \right).[13] This form was described and popularized by Emil J. Gumbel in his seminal 1958 work Statistics of Extremes, which focused on the distribution of maxima from normal distributions, and the GEV framework later unified it with other extreme value types.[14] The Gumbel distribution features tails with double exponential decay on the right side, which aligns well with extremes from parent distributions exhibiting light tails, such as exponential or subexponential decay (e.g., normal or lognormal).[15] This property distinguishes it from heavier-tailed cases in the GEV family.[15] The standardized form of the Gumbel distribution sets μ = 0 and σ = 1, resulting in the CDF F(z) = \exp( -\exp( -z ) ) and PDF f(z) = \exp( -z - \exp( -z ) ), where z is the standardized variable.[13] This reduction facilitates comparisons and simulations in extreme value analysis.[13] In practice, the Gumbel distribution (with ξ ≈ 0) is commonly applied to model annual maximum flood discharges in hydrology, enabling estimation of return periods for flood events based on historical peak flow data.[16]Fréchet distribution case
The Fréchet case of the generalized extreme value (GEV) distribution arises when the shape parameter \xi > 0, resulting in a heavy right tail that allows for the possibility of extremely large values. In this scenario, the cumulative distribution function (CDF) of the GEV is given by G(z; \mu, \sigma, \xi) = \exp\left\{ -\left[1 + \xi \frac{z - \mu}{\sigma}\right]^{-1/\xi} \right\}, defined for $1 + \xi (z - \mu)/\sigma > 0, where the term \left[1 + \xi (z - \mu)/\sigma\right]^{-1/\xi} dominates the behavior for large z, leading to a slowly decaying tail probability. This case connects directly to the classical Fréchet distribution, which serves as the limiting form for maxima of sequences from distributions with power-law tails. The standard Fréchet distribution, with shape parameter \alpha > 0, has CDF \Phi_\alpha(z) = \exp\left(-z^{-\alpha}\right), \quad z > 0, and in the GEV parameterization, \alpha = 1/\xi, unifying the heavy-tailed extremes under a single framework. This distribution was first introduced by Maurice Fréchet in 1927.[17] The support of the distribution in the Fréchet case is z > \mu - \sigma/\xi, which is bounded below but unbounded above, reflecting the potential for arbitrarily large maxima without an upper limit. Physically, the Fréchet case models block maxima from underlying distributions exhibiting power-law decay in their tails, such as the Pareto distribution, where extreme events occur with probabilities that decrease polynomially rather than exponentially. Representative applications include the analysis of stock market crashes, where heavy-tailed losses can be captured to assess tail risk,[18] and earthquake magnitudes, which follow power-law behaviors suitable for Fréchet modeling of maximum seismic events.[19]Weibull distribution case
When the shape parameter \xi < 0, the generalized extreme value (GEV) distribution corresponds to the Weibull case, characterized by a finite upper endpoint at \mu - \sigma / \xi, where \mu is the location parameter and \sigma > 0 is the scale parameter. This configuration arises in the limiting distribution of block maxima from parent distributions with a hard upper bound, such as uniform or beta distributions, where extremes cannot exceed a physical or theoretical limit. The support of the distribution is restricted to z < \mu - \sigma / \xi, ensuring all probability mass lies below this endpoint. This Weibull case relates directly to the reversed Weibull distribution, which models maxima rather than minima; the standard Weibull is typically used for minima of distributions bounded below, whereas the reversed form applies to upper-bounded maxima. The Type III extreme value distribution was identified by Fisher and Tippett in 1928, with the Weibull distribution parameterized for such extremes by Waloddi Weibull in 1939.[5] The shape parameter \alpha of the reversed Weibull is given by \alpha = -1 / \xi > 0, linking the tail behavior to the GEV's extremal type.[20] Near the upper endpoint, the density function decays rapidly, reflecting the abrupt cutoff in the parent distribution's tail, which makes this case appropriate for phenomena like material strength failures or bounded environmental extremes. Conventions in the literature vary, with some sources referring to the \xi < 0 GEV simply as the Weibull extreme value distribution, while others emphasize the "reversed" aspect to distinguish it from the minima-oriented Weibull and avoid confusion with the positive-shape Fréchet case.[21] This nomenclature traces back to early formulations in extreme value theory, where the three types (Gumbel, Fréchet, and Weibull) were unified under the GEV framework.Properties
Moments and summary statistics
The moments of the generalized extreme value (GEV) distribution, denoted as Z \sim \text{GEV}(\mu, \sigma, \xi), exist only under certain conditions on the shape parameter \xi. The mean exists for \xi < 1, the variance for \xi < 1/2, the third central moment (and thus skewness) for \xi < 1/3, and the fourth central moment (and thus kurtosis) for \xi < 1/4. These conditions arise because higher moments involve gamma functions \Gamma(1 - k\xi) that diverge as \xi approaches or exceeds $1/k from below. For \xi \geq 0, the distribution is unbounded above, leading to heavy tails that cause moment divergence at these thresholds; for \xi < 0, the distribution is bounded above at \mu - \sigma / \xi, but moments still fail beyond the specified limits due to the shape of the density.[22] The mean is given by E[Z] = \mu + \frac{\sigma}{\xi} \left[ \Gamma(1 - \xi) - 1 \right] for \xi \neq 0 and \xi < 1, where \Gamma denotes the gamma function. When \xi = 0, the distribution reduces to the Gumbel case, and the mean simplifies to E[Z] = \mu + \sigma \gamma, with \gamma \approx 0.57721 the Euler-Mascheroni constant. For \xi \geq 1, the mean is infinite. The location parameter \mu shifts the mean directly, while the scale \sigma > 0 stretches it proportionally, and \xi controls the tail heaviness that affects the gamma term.[23] The variance is \text{Var}(Z) = \frac{\sigma^2}{\xi^2} \left[ \Gamma(1 - 2\xi) - \left[ \Gamma(1 - \xi) \right]^2 \right] for \xi \neq 0 and \xi < 1/2. For \xi = 0, it becomes \text{Var}(Z) = \frac{\pi^2}{6} \sigma^2 \approx 1.64493 \sigma^2. The variance is infinite for \xi \geq 1/2. As with the mean, \mu does not affect the variance, but \sigma scales it quadratically, and small deviations of \xi from zero introduce asymmetry that alters the spread.[23] The skewness, a measure of asymmetry, is \gamma_1 = \text{sgn}(\xi) \frac{ \Gamma(1 - 3\xi) - 3 \Gamma(1 - 2\xi) \Gamma(1 - \xi) + 2 \left[ \Gamma(1 - \xi) \right]^3 }{ \left\{ \Gamma(1 - 2\xi) - \left[ \Gamma(1 - \xi) \right]^2 \right\}^{3/2} } for \xi \neq 0 and \xi < 1/3, scaled by \sigma^{-3} times the third central moment expression but normalized; it is infinite for \xi \geq 1/3. For \xi = 0, \gamma_1 \approx 1.13955 > 0, indicating right-skewness typical of extreme value maxima. For small \xi, skewness approximates $1.14 (1 + 6 \xi^2 + O(\xi^3)), showing mild sensitivity to \xi near the Gumbel limit. Positive \xi enhances right-skewness (heavy upper tail), while negative \xi induces left-skewness (bounded upper tail).[22] The (excess) kurtosis, measuring tail heaviness relative to the normal distribution, is \gamma_2 = \frac{ \Gamma(1 - 4\xi) - 4 \Gamma(1 - 3\xi) \Gamma(1 - \xi) + 6 \Gamma(1 - 2\xi) \left[ \Gamma(1 - \xi) \right]^2 - 3 \left[ \Gamma(1 - \xi) \right]^4 }{ \left\{ \Gamma(1 - 2\xi) - \left[ \Gamma(1 - \xi) \right]^2 \right\}^2 } - 3 for \xi \neq 0 and \xi < 1/4, with the unnormalized fourth central moment scaled by \sigma^{-4}; it diverges for \xi \geq 1/4. For \xi = 0, excess kurtosis is $12/5 = 2.4, so total kurtosis is 5.4, reflecting leptokurtic tails compared to the normal (kurtosis 3). For small \xi, it approximates $2.4 (1 + 24 \xi^2 + O(\xi^3)), increasing tail thickness with |\xi|. This leptokurtosis underscores the GEV's suitability for modeling extremes, where outliers are more probable than under Gaussian assumptions.[22] The median, as the 50th percentile, is m = \mu + \frac{\sigma}{\xi} \left[ (\ln 2)^{-\xi} - 1 \right] for \xi \neq 0, and for \xi = 0, m = \mu - \sigma \ln (-\ln 0.5) = \mu - \sigma \ln (\ln 2). This places the median below the mean for \xi > 0 due to right-skewness and above for \xi < 0. The mode, the value maximizing the probability density function, exists for \xi < 1 and is \text{mode} = \mu + \frac{\sigma}{\xi} \left[ (1 + \xi)^{-\xi} - 1 \right]. For \xi = 0, the mode coincides with \mu. The mode shifts left of the median and mean for \xi > 0, emphasizing the peaked, skewed nature of the density near extremes.Quantile function and mode
The quantile function of the generalized extreme value (GEV) distribution provides the inverse of the cumulative distribution function, enabling the computation of values corresponding to specific probabilities, which is essential for simulating extremes and determining return levels in applications such as flood risk assessment. For the case \xi \neq 0, the quantile function is Q(p) = \mu + \frac{\sigma}{\xi} \left[ (-\ln p)^{-\xi} - 1 \right], \quad p \in (0,1), provided that $1 + \xi (Q(p) - \mu)/\sigma > 0.[24] When \xi = 0, it reduces to the Gumbel case: Q(p) = \mu - \sigma \ln(-\ln p), \quad p \in (0,1)./05%3A_Special_Distributions/5.30%3A_The_Extreme_Value_Distribution) These expressions facilitate generating random variates from the GEV distribution by applying the inverse transform sampling method and are particularly valuable for peak analysis in time series of maxima.[25] The mode of the GEV distribution, representing the most probable value and thus the peak of the density, is obtained by maximizing the probability density function. The standard GEV PDF is g(x) = \frac{1}{\sigma} \left[1 + \xi \frac{x - \mu}{\sigma} \right]^{-1/\xi - 1} \exp\left( -\left[1 + \xi \frac{x - \mu}{\sigma} \right]^{-1/\xi} \right). Setting the derivative of the log-PDF with respect to x to zero and solving yields, for \xi \neq 0, m = \mu + \frac{\sigma}{\xi} \left[ (1 + \xi)^{-\xi} - 1 \right], subject to the support condition $1 + \xi (m - \mu)/\sigma > 0. For \xi = 0, the mode simplifies to m = \mu.[25] This location of the mode provides insight into the central tendency of extreme observations and aids in interpreting the shape parameter's influence on the distribution's peak. The GEV distribution is unimodal across all valid parameter combinations, ensuring a single maximum in the density for reliable peak identification in data analysis.[25]Transformation for minima
In extreme value theory, the generalized extreme value (GEV) distribution is typically formulated to model block maxima, but it can be readily adapted to model block minima through a negation transformation. Consider a sequence of independent and identically distributed random variables X_1, \dots, X_n. The block minimum is defined as M_n = \min(X_1, \dots, X_n). This can be expressed as M_n = -\max(-X_1, \dots, -X_n), linking the minima of the X_i to the maxima of the negated variables -X_i. If the normalized maxima of the -X_i converge to a GEV distribution with location parameter \mu, scale parameter \sigma > 0, and shape parameter \xi, then the normalized minima M_n follow a GEV distribution with location -\mu', scale \sigma', and shape -\xi, where \mu' and \sigma' are appropriately chosen normalizing constants related to the original series. This transformation preserves the form of the GEV cumulative distribution function (CDF). The standard GEV CDF for maxima is H(x; \mu, \sigma, \xi) = \exp\left\{ -\left[1 + \xi \frac{x - \mu}{\sigma}\right]_+^{-1/\xi} \right\}, defined for $1 + \xi (x - \mu)/\sigma > 0. For the minima, the CDF becomes P(M_n \leq z) \approx H(-z; -\mu, \sigma, -\xi), which simplifies to \exp\left\{ -\left[1 - \xi \frac{z + \mu}{\sigma}\right]_+^{1/\xi} \right\}, again defined on the appropriate support. The rationale for this symmetry lies in the duality of extreme value limits: the class of distributions attracting to the GEV for maxima is closed under negation, allowing minima to be handled equivalently by flipping the sign of the observations and adjusting parameters accordingly. This approach avoids deriving a separate limiting form and leverages established theory for maxima. The support of the transformed GEV adjusts based on the shape parameter. For the original maxima of -X_i with \xi > 0 (Fréchet case, heavy-tailed upper extremes), the minima of X_i have shape -\xi < 0 (Weibull case, bounded lower endpoint at -\mu - \sigma/\xi), but if the original \xi < 0, the minima exhibit an unbounded lower tail suitable for modeling extremes without a finite lower bound, such as deep lows in unbounded processes. Conversely, for \xi = 0, the transformation yields the Gumbel case for minima with exponential lower tail. These adjustments ensure the model captures the tail behavior appropriate for low extremes. A practical example is the modeling of annual minimum temperatures, where negation transforms the coldest values into largest magnitudes for fitting a standard GEV to maxima. For instance, analysis of minimum temperatures in northern Sweden used this approach to estimate return levels for extreme cold events, confirming the transformation's utility in climate applications. Similarly, in hydrology, annual minimum streamflows (low flows) are modeled by negating the series and fitting a GEV to the resulting maxima, enabling frequency analysis of drought severity.[26]Parameter Estimation and Inference
Maximum likelihood estimation
Maximum likelihood estimation (MLE) for the parameters of the generalized extreme value (GEV) distribution involves maximizing the likelihood function based on a sample of independent observations assumed to follow the GEV density. For a sample z_1, \dots, z_n, the likelihood is given by L(\mu, \sigma, \xi \mid \mathbf{z}) = \prod_{i=1}^n g(z_i; \mu, \sigma, \xi), where g(\cdot; \mu, \sigma, \xi) denotes the GEV probability density function, and the log-likelihood is l(\mu, \sigma, \xi) = \sum_{i=1}^n \log g(z_i; \mu, \sigma, \xi). [27] No closed-form expressions exist for the maximum likelihood estimators (MLEs) of the location parameter \mu, scale parameter \sigma > 0, and shape parameter \xi, so numerical optimization techniques, such as Newton-Raphson or quasi-Newton methods, are employed to maximize the log-likelihood.[28] These methods can face challenges, particularly when \xi approaches boundary values (e.g., near 0 or negative thresholds where the support changes), leading to potential instability or convergence issues in finite samples.[29] Under standard regularity conditions, the MLEs are consistent and asymptotically normally distributed as the sample size n \to \infty, with the asymptotic covariance matrix derived from the inverse of the observed or expected Fisher information matrix.[30] The information matrix for the GEV parameters has explicit expressions that facilitate standard error estimation.[27] In practice, MLE for the GEV is implemented in statistical software, such as theextRemes package in R, which uses the fevd function to fit the model via numerical maximization, and in Python's scipy.stats module via the genextreme.fit method.[31][32] Due to small-sample bias in the shape parameter \xi, particularly for moderate n, bias correction techniques are recommended; profile likelihood methods, which maximize the log-likelihood over \mu and \sigma for fixed \xi, provide a basis for bias-reduced estimation and inference on \xi.[33]