Fact-checked by Grok 2 weeks ago

Gumbel distribution

The Gumbel distribution, also known as the type I extreme value distribution, is a continuous probability distribution that models the limiting distribution of the maximum (or the negative of the minimum) of a large number of independent and identically distributed random variables drawn from distributions in its domain of attraction, such as the exponential, normal, lognormal, or gamma distributions.^[1]^[2] It plays a central role in extreme value theory as one of three asymptotic forms for maxima, specifically the Fisher-Tippett type I, which applies to distributions with exponentially decaying tails.^[1]^[2] The distribution is parameterized by a location parameter μ (real-valued) and a scale parameter β > 0.^[1]^[2] For the maximum case, the probability density function (PDF) is given by
f(x; \mu, \beta) = \frac{1}{\beta} \exp\left( -\frac{x - \mu}{\beta} - \exp\left( -\frac{x - \mu}{\beta} \right) \right),
and the cumulative distribution function (CDF) is
F(x; \mu, \beta) = \exp\left( -\exp\left( -\frac{x - \mu}{\beta} \right) \right). ^[2]^[1] The minimum case uses a reflected form with PDF
f(x; \mu, \beta) = \frac{1}{\beta} \exp\left( \frac{x - \mu}{\beta} - \exp\left( \frac{x - \mu}{\beta} \right) \right)
and CDF
F(x; \mu, \beta) = 1 - \exp\left( -\exp\left( \frac{x - \mu}{\beta} \right) \right). ^[2] In the standard form, μ = 0 and β = 1.^[2] Notable properties include a mean of μ + γβ for the maximum case, where γ ≈ 0.57721 is the Euler-Mascheroni constant, and a variance of (π²/6)β² ≈ 1.64493β².^[1]^[2] The skewness is positive at approximately 1.13955, and the kurtosis is 5.4 (excess kurtosis of 2.4), reflecting its asymmetric tail toward higher values.^[2] The mode is at μ, and the median is μ - β ln(ln 2) ≈ μ + 0.366513β.^[2] The distribution is stable under maxima operations, meaning the maximum of independent Gumbel variables follows a shifted and scaled Gumbel distribution.^[1] Originally derived by Ronald A. Fisher and Leonard H.C. Tippett in 1928 as part of their classification of extreme value limits, the distribution gained prominence through the applications of Emil J. Gumbel, who first used it systematically for flood frequency analysis in 1941 and detailed its theory in his 1958 book Statistics of Extremes.^[1]^[3] It forms a special case of the generalized extreme value (GEV) distribution when the shape parameter is zero.^[2] The Gumbel distribution finds extensive use in fields requiring modeling of rare events, such as hydrology for predicting flood peaks and rainfall extremes, reliability engineering for assessing material strengths and failure times, meteorology for wind speed maxima, and environmental science for analyzing pollutant concentrations.^[2]^[3] In these applications, it enables estimation of return levels for events with specified probabilities, often fitted via methods like probability-weighted moments or maximum likelihood.^[2]

Definitions

Standard Gumbel Distribution

The standard Gumbel distribution is a continuous probability distribution that serves as the limiting form for the normalized maxima of a large number of independent and identically distributed random variables drawn from distributions with exponentially decaying tails, such as the normal or exponential distributions.^[4] This distribution, also known as the Type I extreme value distribution, was originally identified in the foundational work on extreme value theory.^[5] The probability density function (PDF) of the standard Gumbel distribution is defined as

f(x) = e^{-x} \exp\left(-e^{-x}\right), \quad x \in \mathbb{R}.

^[4] Its cumulative distribution function (CDF) is

F(x) = \exp\left(-e^{-x}\right), \quad x \in \mathbb{R}.

^[4] A sketch of its derivation begins with considering the sample maximum M_n = \max\{X_1, \dots, X_n\} from i.i.d. random variables X_i with CDF G(x) in the Gumbel domain of attraction; suitable normalizing constants a_n > 0 and b_n are chosen such that \lim_{n \to \infty} P((M_n - b_n)/a_n \leq x) = F(x), yielding the standard Gumbel CDF as the non-degenerate limit.^[4]^[5] The distribution is located at its mode of x = 0 and features an asymmetric, right-skewed shape with a heavier tail on the positive side.^[4]

Generalized Gumbel Distribution

The generalized Gumbel distribution extends the standard form into a location-scale family, enabling it to model shifted and rescaled extreme values observed in various datasets. It is parameterized by a location parameter \mu \in \mathbb{R}, which shifts the distribution along the real line, and a scale parameter \beta > 0, which stretches or compresses it to adjust for variability in the data. This parameterization arises naturally in extreme value theory, where the limiting distribution of normalized maxima or minima from many underlying distributions requires such flexibility for practical application.^[2] The probability density function (PDF) of the generalized Gumbel distribution is

f(x; \mu, \beta) = \frac{1}{\beta} \exp\left( -\frac{x - \mu}{\beta} \right) \exp\left( -\exp\left( -\frac{x - \mu}{\beta} \right) \right)

for x \in \mathbb{R}.^[2] The corresponding cumulative distribution function (CDF) is

F(x; \mu, \beta) = \exp\left( -\exp\left( -\frac{x - \mu}{\beta} \right) \right)

for x \in \mathbb{R}.^[2] The location parameter \mu serves as the mode of the distribution, representing the peak of the PDF.^[6] Meanwhile, the scale parameter \beta governs the distribution's spread and degree of right-skewed asymmetry, with larger values of \beta increasing both the tail heaviness on the right and the overall dispersion.^[2] The survival function, which gives the probability of exceeding a threshold, is

S(x; \mu, \beta) = 1 - F(x; \mu, \beta) = 1 - \exp\left( -\exp\left( -\frac{x - \mu}{\beta} \right) \right).

^[2] This form highlights the distribution's utility in reliability and risk assessment, where tail probabilities are of primary interest.

Properties

Moments and Central Tendency

The Gumbel distribution, parameterized by location \mu and scale \beta > 0, exhibits specific measures of central tendency that reflect its asymmetric nature. The mode, which corresponds to the peak of the probability density function, occurs exactly at the location parameter \mu.^[2] The median m, defined as the value where the cumulative distribution function equals 0.5, is given by m = \mu - \beta \ln(\ln 2), which numerically approximates to \mu + 0.36651 \beta since \ln(\ln 2) \approx -0.36651.^[2] The mean, or expected value E[X], is \mu + \beta \gamma, where \gamma \approx 0.57721 is the Euler-Mascheroni constant; this positions the mean to the right of both the mode and median, consistent with the distribution's positive skewness.^[1]^[7] Higher moments provide further insight into the distribution's spread and shape. The variance \mathrm{Var}(X) is \frac{\pi^2}{6} \beta^2, scaling quadratically with the scale parameter and determining the overall dispersion around the mean.^[2]^[1] The skewness \gamma_1, measuring asymmetry, is \frac{12 \sqrt{6} \zeta(3)}{\pi^3} \approx 1.13955, where \zeta(3) \approx 1.20206 is Apéry's constant; this positive value indicates a right-tailed asymmetry, with the tail extending further on the positive side.^[1]^[7] The excess kurtosis \gamma_2 is \frac{12}{5} = 2.4, signifying heavier tails than the normal distribution (which has excess kurtosis of 0) and a total kurtosis of 5.4; this leptokurtic property underscores the distribution's proneness to extreme outliers.^[1]^[2] These moments collectively summarize the central behavior of the Gumbel distribution, with the scale \beta influencing both location and spread while \mu shifts the entire profile. In practice, the mean serves as a primary summary statistic for location in symmetric approximations, though the median may be preferred for skewed data to mitigate tail influence.^[7]

Shape and Scale Characteristics

The Gumbel distribution exhibits a right-skewed bell-shaped probability density function, characterized by a sharp decline on the left side and a more gradual exponential decay on the right tail, making it particularly suitable for modeling the distribution of extreme maxima.^[8]^[9] This asymmetry arises from its role in extreme value theory, where the density function is given by f(x) = \frac{1}{\beta} \exp\left( -\frac{x - \mu}{\beta} - \exp\left( -\frac{x - \mu}{\beta} \right) \right) for the standard form used for maxima, with \mu as the location parameter and \beta > 0 as the scale parameter.^[2] The tail behavior further emphasizes this skewness: the left tail of the cumulative distribution function satisfies F(x) \sim \exp\left( -\exp\left( \frac{\mu - x}{\beta} \right) \right) as x \to -\infty, indicating double-exponential decay, while the right tail follows $1 - F(x) \sim \exp\left( -\frac{x - \mu}{\beta} \right) as x \to \infty, reflecting exponential decay that allows for heavier extreme values on the upper end.^[8] These asymptotics highlight the distribution's ability to capture unbounded upper extremes without a finite endpoint, contrasting with lighter-tailed alternatives. The quantile function, or inverse cumulative distribution function, is explicitly Q(p) = \mu - \beta \ln(-\ln p) for p \in (0,1), providing a direct way to compute thresholds for given probabilities and underscoring the distribution's utility in risk assessment.^[2] The location parameter \mu shifts the entire distribution along the real line without altering its shape, while the scale parameter \beta controls the spread: larger values of \beta widen the distribution, increasing the variance and extending both tails proportionally.^[2] Compared to the normal distribution, the Gumbel features a heavier right tail due to its exponential decay, enabling better modeling of rare large events, but a lighter left tail owing to the sharper double-exponential drop-off, resulting in less probability mass in the lower extremes.^[8] This tail disparity contributes to its positive skewness, approximately 1.14, distinguishing it from the symmetric normal.^[2]

Connections to Extreme Value Theory

The Gumbel distribution plays a central role in extreme value theory (EVT) as the limiting distribution for the maxima of sequences of independent and identically distributed (i.i.d.) random variables drawn from parent distributions in its domain of attraction. It serves as the Type I extreme value distribution, applicable to maxima from distributions exhibiting exponentially decaying tails, including the exponential, normal, and lognormal distributions. The Fisher–Tippett–Gnedenko theorem establishes that the possible non-degenerate limiting distributions for normalized sample maxima fall into three types, with the Gumbel distribution emerging when the tail index ξ = 0 in the generalized extreme value (GEV) distribution.^[10] This theorem implies that, for i.i.d. random variables X_1, X_2, \dots, X_n with cumulative distribution function F and M_n = \max\{X_1, \dots, X_n\}, there exist normalizing constants a_n > 0 and b_n such that

\lim_{n \to \infty} P\left( \frac{M_n - b_n}{a_n} \leq x \right) = G(x) = \exp\left( -\exp(-x) \right), \quad -\infty < x < \infty,

provided F belongs to the Gumbel domain of attraction.^[11] Distributions in this domain are characterized by tails that decay exponentially, satisfying conditions such as \lim_{t \to \infty} \frac{1 - F(t + x \gamma(t))}{1 - F(t)} = e^{-x} for an auxiliary function \gamma(t) > 0 and x \in \mathbb{R}, which ensures convergence to the Gumbel limit.^[11] Historically, the theorem originated with the 1928 work of Ronald A. Fisher and Leonard H. C. Tippett, who derived possible limiting forms for extreme order statistics, and was fully proven by Boris V. Gnedenko in 1943, providing the rigorous classification into three types.^[5]^[12] The distribution is named after Emil J. Gumbel, whose applications to flood analysis starting in the 1940s built on this foundation and popularized its role in EVT.^[13]

Transformations and Variants

The Gumbel distribution exhibits several important transformations that connect it to other distributions in extreme value theory. One key transformation involves the exponential function: if Y \sim Gumbel(\mu, \beta), then e^Y follows a Fréchet distribution with shape parameter $1/\beta and scale e^\mu.^[14] This relation arises because the logarithmic transformation of Fréchet variates yields Gumbel variates, facilitating analysis of heavy-tailed extremes through lighter-tailed equivalents.^[14] For modeling minima rather than maxima, the Gumbel distribution can be reflected. If Y \sim Gumbel(\mu, \beta) for maxima, then -Y follows a Gumbel distribution for minima with parameters -\mu, \beta. The probability density function for the minima case is given by

f(x; \mu, \beta) = \frac{1}{\beta} \exp\left( \frac{x - \mu}{\beta} - \exp\left( \frac{x - \mu}{\beta} \right) \right),

which mirrors the standard form for maxima by flipping the signs in the exponents.^[15] A notable relation exists between the Gumbel and the logistic distribution: the difference of two independent Gumbel random variables with the same scale parameter follows a logistic distribution. Specifically, if X \sim Gumbel(\mu_1, \beta) and Y \sim Gumbel(\mu_2, \beta), then X - Y \sim Logistic(\mu_1 - \mu_2, \beta).^[16] This property underpins applications in choice modeling and logit analysis, where Gumbel errors lead to logistic differences.^[17] The Gumbel distribution also admits a discrete analog for approximating integer-valued extremes. The discrete Gumbel distribution is derived by differencing the cumulative distribution function (CDF) of the continuous Gumbel, yielding a probability mass function suitable for count data in extreme value contexts. For a discrete random variable K taking integer values, the PMF is P(K = k) = F(k) - F(k-1), where F is the continuous Gumbel CDF, providing a bridge between continuous and discrete extreme value modeling. Finally, the Gumbel distribution emerges as a special case of the generalized extreme value (GEV) distribution when the shape parameter \xi = 0. In the GEV family, which unifies the three types of extreme value distributions, the limiting form as \xi \to 0 recovers the Gumbel, distinguishing it from the Fréchet (\xi > 0) and reversed Weibull (\xi < 0) cases.^[18] This boundary role positions the Gumbel as the exponential-tailed member within the broader GEV framework.

Applications

Extreme Value Analysis

The block maxima approach in extreme value analysis involves selecting the maximum value from non-overlapping blocks of data, such as annual maxima from daily observations, and fitting a Gumbel distribution to model these long-term extremes under stationarity.^[19]^[20] This method leverages the fact that, for many underlying distributions (e.g., normal or exponential), the normalized block maxima converge to the Gumbel distribution as a special case of the generalized extreme value (GEV) distribution with shape parameter ξ = 0.^[19]^[21] The cumulative distribution function (CDF) of the Gumbel distribution for maxima is given by

F(x) = \exp\left(-\exp\left(-\frac{x - \mu}{\beta}\right)\right),

where μ is the location parameter and β > 0 is the scale parameter, enabling straightforward modeling of the tail behavior of rare events.^[2] Return levels, which quantify the magnitude of an extreme event expected to occur once every T periods (with exceedance probability 1/T), are derived directly from the Gumbel CDF. The T-year return level is

x_T = \mu - \beta \ln\left(-\ln\left(1 - \frac{1}{T}\right)\right).

This closed-form expression facilitates probabilistic forecasting of events like floods or high winds, providing engineers with design thresholds for infrastructure resilience. While the peaks-over-threshold (POT) method serves as an alternative by modeling exceedances above a high threshold using the generalized Pareto distribution, the block maxima approach with the Gumbel distribution is often preferred for directly capturing the distribution of exact maxima in long-term records, avoiding threshold selection biases.^[22] In hydrological applications, such as river flood modeling, the Gumbel distribution has been applied since the 1940s, following Emil Julius Gumbel's early forecasting work on flood peaks in the United States.^[23] Similar uses extend to estimating extreme wind speeds for structural design and analyzing failure times in reliability engineering, where the Gumbel models minima of lifetimes (equivalent to maxima of negative values).^[24]^[25] A key advantage of the Gumbel distribution in extreme value analysis is its closed-form CDF, which simplifies the computation of confidence intervals for return levels and parameter estimates, enhancing uncertainty quantification in risk assessments.^[26] For instance, resampling techniques applied to Gumbel-fitted block maxima yield reliable interval estimates for hydrological extremes, supporting informed decision-making in flood management.^[26]

Statistical Modeling and Prediction

In survival analysis, the Gumbel distribution is employed to model extremes such as lifetime maxima or censoring thresholds, particularly in scenarios involving right-censored data where observations are truncated due to study endpoints or competing risks.^[27] This approach accommodates the asymptotic behavior of maxima from underlying distributions in the Gumbel domain of attraction, enabling robust inference on tail events even with incomplete observations.^[28] For instance, it facilitates estimation of extreme survival times by treating censored values as upper bounds, which is crucial for reliability engineering and clinical trials assessing long-term outcomes.^[29] Bayesian applications leverage the Gumbel distribution within hierarchical models to incorporate location-scale parameters, often using non-informative or weakly informative priors to update beliefs about extreme quantiles across groups.^[30] This setup allows for pooling information in multilevel structures, such as varying scale parameters for subpopulations, enhancing predictive accuracy in settings like environmental monitoring or risk assessment.^[31] Although not strictly conjugate in the exponential family sense, the distribution's location-scale invariance supports efficient posterior sampling via Markov chain Monte Carlo methods in these frameworks.^[32] In machine learning, the Gumbel-softmax trick reparameterizes the softmax function by adding Gumbel noise to logits, enabling differentiable sampling from categorical distributions for multi-class probability estimation during training.^[33] This technique addresses the non-differentiability of discrete choices in neural networks, facilitating backpropagation in variational autoencoders and reinforcement learning agents that require stochastic decision-making.^[34] It approximates the categorical distribution while maintaining low-variance gradients, improving convergence in tasks like text generation or policy optimization.^[35] The Gumbel distribution's quantile function is instrumental in constructing prediction intervals for forecasting extreme events, such as stock market crashes, by specifying upper bounds on maximum losses with specified confidence levels.^[36] In finance, this supports value-at-risk computations, where the inverse cumulative distribution provides thresholds for tail risks in return series modeled under extreme value assumptions.^[37] Such intervals quantify uncertainty in downturn predictions, aiding portfolio stress testing and regulatory compliance.^[38] Software implementations facilitate these applications, with the R package evd providing functions for Gumbel density, quantile, and random generation tailored to extreme value modeling.^[39] In Python, scipy.stats.gumbel_r offers a right-skewed Gumbel class with methods for parameter fitting and interval estimation, integrable into predictive pipelines.^[40] A discrete analog of the Gumbel distribution extends these tools to count-based predictions, such as daily survival counts.^[41]

Parameter Estimation

Method of Moments

The method of moments estimation for the Gumbel distribution parameters relies on equating the first two sample moments to the population moments, providing a straightforward approach to parameter fitting. The population mean is \mu + \beta \gamma, where \gamma \approx 0.57721 is the Euler-Mascheroni constant, and the population variance is \frac{\pi^2 \beta^2}{6}.^[42]^[2] To apply this method, first compute the sample mean \bar{X} and the sample standard deviation S from the observed data. The scale parameter estimator is then given by

\hat{\beta} = \frac{S}{\pi / \sqrt{6}},

and the location parameter estimator by

\hat{\mu} = \bar{X} - \hat{\beta} \gamma.

These closed-form expressions allow for direct calculation without iterative optimization.^[2]^[42] This estimation technique offers advantages in simplicity and computational efficiency, as it yields explicit formulas that are easy to implement, particularly for small to moderate sample sizes where more complex methods may struggle.^[43] The estimators are asymptotically unbiased and consistent, converging to the true parameters as the sample size increases. However, they display finite-sample bias, especially in \hat{\beta}, which can affect accuracy in smaller datasets.^[44]^[43] Compared to maximum likelihood estimation, the method of moments is less statistically efficient for large samples, as evidenced by higher mean squared errors in simulation studies, though it remains a viable initial approximation.^[44]^[43]

Maximum Likelihood Estimation

The maximum likelihood estimator (MLE) for the parameters of the Gumbel distribution, location μ and scale β > 0, is obtained by maximizing the log-likelihood function based on an independent and identically distributed sample x_1, \dots, x_n. The log-likelihood is given by

\ell(\mu, \beta) = -n \ln \beta - \sum_{i=1}^n \frac{x_i - \mu}{\beta} - \sum_{i=1}^n \exp\left( -\frac{x_i - \mu}{\beta} \right).

^[42] To find the MLEs, the score equations are solved by setting the partial derivatives to zero:

\frac{\partial \ell}{\partial \mu} = \frac{1}{\beta} \left( n - \sum_{i=1}^n \exp\left( -\frac{x_i - \mu}{\beta} \right) \right) = 0,

\frac{\partial \ell}{\partial \beta} = \frac{1}{\beta} \sum_{i=1}^n \frac{x_i - \mu}{\beta} \left( 1 - \exp\left( -\frac{x_i - \mu}{\beta} \right) \right) - \frac{n}{\beta} = 0.

The second equation can be rewritten using the solution from the first, but it involves terms that require the digamma function in expected value calculations for approximations.^[42] There is no closed-form solution for the MLEs, so they must be obtained iteratively using numerical methods such as Newton-Raphson, which updates parameter estimates via the Hessian matrix of second derivatives until convergence.^[45] Under standard regularity conditions, the MLEs \hat{\mu} and \hat{\beta} are asymptotically normal with mean (\mu, \beta) and covariance matrix given by the inverse of the Fisher information matrix scaled by $1/n. The Fisher information matrix per observation has elements

I_{11} = \frac{\pi^2}{6 \beta^2}, \quad I_{12} = I_{21} = 0, \quad I_{22} = \frac{1 + \frac{\pi^2}{6} - \gamma^2}{\beta^2},

where \gamma \approx 0.57721 is the Euler-Mascheroni constant. This yields asymptotic variances \operatorname{Var}(\hat{\mu}) \approx \frac{6 \beta^2}{\pi^2 n} and \operatorname{Var}(\hat{\beta}) \approx \frac{\beta^2}{(1 + \frac{\pi^2}{6} - \gamma^2) n}.^[46] Model diagnostics for the fitted Gumbel distribution include quantile-quantile (Q-Q) plots, which compare the ordered sample values against theoretical Gumbel quantiles \mu + \beta \ln(-\ln(1 - p_i)) for p_i = (i - 0.5)/n, to assess goodness-of-fit visually. Deviations from linearity indicate potential lack of fit.^[2]

Random Variate Generation

Inversion and Simulation Methods

The inversion method provides a direct and efficient approach to generating random variates from the Gumbel distribution due to its closed-form quantile function. To generate a single variate X with location parameter \mu and scale parameter \beta > 0, first draw U \sim \text{[Uniform](/page/Uniform)}(0,1), then compute

X = \mu - \beta \ln(-\ln U).

This transformation follows from the inverse cumulative distribution function of the Gumbel, ensuring X follows the target distribution exactly.^[47]^[48] Implementation is straightforward and computationally inexpensive: generate a sequence of independent uniform variates U_1, U_2, \dots, U_n, apply the formula to each to obtain X_1, X_2, \dots, X_n, and use these as the simulated Gumbel samples. The method's efficiency stems from the simplicity of the logarithmic operations and the absence of iterative steps, making it suitable for large-scale simulations.^[48] To validate generated samples, compute empirical moments (e.g., sample mean and variance) and compare them to theoretical values: the mean is \mu + \beta \gamma (where \gamma \approx 0.57721 is the Euler-Mascheroni constant) and the variance is \beta^2 \pi^2 / 6. Monte Carlo experiments confirm that well-generated samples reproduce these moments closely, with deviations decreasing as sample size increases.^[49]^[4]

Reparameterization Techniques

Reparameterization techniques for the Gumbel distribution are essential in machine learning to enable differentiable sampling, allowing gradients to propagate through stochastic nodes during optimization. A key method expresses a Gumbel random variable in terms of uniform noise: for X \sim \Gumbel(\mu, \beta), one can sample X = \mu + \beta (-\ln(-\ln U)), where U \sim \Uniform(0,1). This inversion-based form separates the randomness into the uniform variable U, whose samples are fixed during the backward pass, permitting low-variance gradients via the reparameterization trick.^[50] The Gumbel-Softmax extends this to categorical distributions, providing a continuous surrogate for discrete sampling that supports backpropagation. Given class probabilities \pi = (\pi_1, \dots, \pi_K), the relaxation is defined as

z_k = \frac{\exp\left( (\log \pi_k + g_k)/\tau \right)}{\sum_{j=1}^K \exp\left( (\log \pi_j + g_j)/\tau \right)},

where g_k \stackrel{\iid}{\sim} \Gumbel(0,1) for k = 1, \dots, K, and \tau > 0 is a temperature parameter controlling smoothness. At high \tau, outputs are nearly uniform; as \tau \to 0, they concentrate on one-hot vectors, approximating the categorical argmax. The g_k are themselves reparameterized using uniforms, ensuring the entire process is differentiable.^[50] These techniques find prominent use in variational autoencoders (VAEs) with discrete latents, where Gumbel-Softmax enables amortized inference by relaxing the posterior over categories, yielding lower test reconstruction losses than baselines on datasets like MNIST (e.g., 101.5 nats versus 105.0 nats for Bernoulli VAEs). In reinforcement learning, they facilitate gradient-based policy optimization for discrete actions, as in extensions of actor-critic methods that relax action selection to improve exploration and training stability.^[50] Compared to score-function estimators like REINFORCE, Gumbel-based reparameterizations yield lower-variance gradients, avoiding the linear scaling of variance with output dimensionality and outperforming on tasks such as structured prediction.^[50] Limitations include approximation bias, as finite \tau produces soft samples deviating from true categoricals, potentially affecting downstream one-hot decisions; additionally, very low \tau can cause exploding gradients and poor convergence, often necessitating annealing schedules.^[50]

References

[1]
Gumbel Distribution -- from Wolfram MathWorld
In this work, the term "Gumbel distribution" is used to refer to the distribution corresponding to a minimum extreme value distribution.
[2]
1.3.6.6.16. Extreme Value Type I Distribution
The case where μ = 0 and β = 1 is called the standard Gumbel distribution. The equation for the standard Gumbel distribution (maximum) reduces to. f ( x ) = e ...Missing: definition | Show results with:definition
[3]
[PDF] The Gumbel hypothesis test for left censored observations using ...
Jan 11, 2011 · The type I extreme value distribution is also known as the. Gumbel distribution since Gumbel (1941, 1958) first applied it to flood frequency ...
[4]
[PDF] Extreme Value Distributions : Theory and Applications - Minerva
It is a book about extreme-value distributions both univariate and multivariate - and their applications, supplemented by an up-to-date extensive bibliography, ...
[5]
Limiting forms of the frequency distribution of the largest or smallest ...
Oct 24, 2008 · The limiting distribution, when n is large, of the greatest or least of a sample of n, must satisfy a functional equation which limits its form to one of two ...
[6]
[PDF] Supplementary Material - Proceedings of Machine Learning Research
Let us start from the Gumbel density function. P(x) = 1 σ exp. − x − µ σ. + e. −x−µ σ where µ is the mode (location) and σ is the scale parameter. Using ...
[7]
[PDF] Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists
to as the Fisher-Tippett distribution (type I), the log-Weibull distribution, or the Gumbel distribution after E. J. Gumbel (1891–1966). The extreme value ...
[8]
[PDF] The Fundamentals of Heavy Tails: Properties, Emergence, and ...
exponential of the Exponential. The second heavy-tailed distribution we ... right-tail of the Gumbel distribution decays exponentially. However, the ...
[9]
Probability Playground: The Gumbel Distribution
The Gumbel distribution is used to model the distribution of the maximum of a sample. It is therefore a type of extreme value distribution (type I), ...Missing: definition | Show results with:definition
[10]
An Introduction to Statistical Modeling of Extreme Values
This book develops both the basic theoretical framework of extreme value models and the statistical inferential techniques for using these models in practice.
[11]
Modelling Extremal Events: for Insurance and Finance - SpringerLink
Available as PDF; Read on any device; Instant download; Own it forever. Buy ... A nice feature of Embrechts et al is an opening 20-page 'reader guideline ...
[12]
Introduction to Gnedenko (1943) On the Limiting Distribution of the ...
Gnedenko's paper was the first mathematically rigorous treatment of the fundamental limit theorems of extreme value theory. In its influence on the.Missing: URL | Show results with:URL
[13]
Statistics of Extremes - Emil Julius Gumbel - Google Books
Author, Emil Julius Gumbel ; Publisher, Columbia University Press, 1958 ; Original from, the University of California ; Digitized, Jan 30, 2009 ; ISBN, 0231929587, ...Missing: URL | Show results with:URL
[14]
Extreme Value (Gumbel) Distribution - Wiley Online Library
Nov 29, 2010 · Logarithmic transformations of extreme value variates of Type II (Fréchet) and Type III (Weibull) correspond to Type I Gumbel variates.
[15]
[PDF] Some Useful Extreme Value Derivations - Chase Abram
Dec 3, 2023 · The Frechet(ϵ) distribution with shape ϵ has the following CDF. Therefore we can log the Frechet variables of interest to get them in Gumbel ...
[16]
a generalized logistic distribution - UNL Digital Commons
A generalized logistic distribution is proposed, based on the fact that the difference of two independent Gumbel-distributed randomvariables has the standard ...<|control11|><|separator|>
[17]
[PDF] 3 Logit
The distribution is also called Gumbel and type I extreme value (and ... The parameter β, which denotes workers' relative prefer- ence for goods and ...
[18]
[PDF] Chapter 12 - Extreme Value Theory
This property can be used to check graphically if a given distribution function belongs or not to the maximum domain of attraction of the Fréchet distribution.<|separator|>
[19]
[PDF] IEOR E4602: Quantitative Risk Management - Extreme Value Theory
The block maxima approach to EVT is concerned with the limiting distribution of Mn. 6 (Section 2). Page 7. The Maximum Domain of Attraction.
[20]
[PDF] EVA Tutorial #1 Block Maxima Approach under Nonstationarity
(1) Traditional Methods/Rationale for Extreme Value Analysis. (2) Max Stability/Extremal Types Theorem. (3) Block Maxima Approach under Stationarity.
[21]
[PDF] An Introduction to Extreme Value Statistics - Richard Grotjahn
Limit as ξ → 0. Gumbel: Exponential: G(z) = exp. − exp. − z − µ σ. H(x)=1 ... GEV distributiononly uses the maximum seasonal temperature per year (819 ...
[22]
A Horse Race between the Block Maxima Method and the Peak ...
Abstract. Classical extreme value statistics consists of two fundamental approaches: the block maxima (BM) method and the peak-over-threshold.
[23]
Gumbel, Emil Julius Gumbel, hydraulics, hydrology, Victor Miguel ...
From 1937 on, first in France and after 1940 in the United States, he became an expert forecaster, first of river floods and later of drought levels and other ...
[24]
[PDF] Extreme Wind Speed Analysis for the Natural Shoreline ...
Apr 29, 2022 · Peak wind speeds for three return levels (2-yr, 10-yr and 100-yr) were estimated. The use of the Gumbel provides a consistent distribution for ...Missing: failure examples
[25]
Extreme Value Distributions
The times to first failure may be represented by the extreme value type I distribution for minimum values, so the Gumbel/SEV distribution can be used for ...
[26]
[PDF] The construction of confidence intervals for frequency analysis using ...
When applied in frequency analysis, resampling techniques can provide estimates of the uncertainties in both distribution parameters and quantile estimates in ...
[27]
Extremes of Censored and Uncensored Lifetimes in Survival Data
Feb 26, 2020 · We assume that the distributions of the T_i^* and U_i are in the domain of attraction of the Gumbel distribution and obtain the asymptotic ...
[28]
On Joint Progressively Censored Gumbel Type-II Distributions - MDPI
The analysis confirms that the Gumbel Type-II distribution with joint progressive censoring provides a flexible and effective model for lifetime data, enabling ...
[29]
[PDF] Estimation of the Cure Rate for Distributions in the Gumbel ... - Lirias
The lifetimes of those individuals will necessarily be censored at the limit of follow-up time. The rest of the population, who will eventually experience the ...
[30]
A Bayesian hierarchical model for estimating the statistical ...
In this paper, we proposed a Bayesian hierarchical model (BHM) for estimating the statistical parameters for monthly average streamflows. We assumed that the ...
[31]
BAYESIAN ANALYSIS OF GUMBEL MODEL BY USING SLICE ...
This study assesses accuracy of SS algorithm and compares it with other techniques such as Metropolis Hasting (MH) algorithm and Maximum Likelihood (ML) in ...
[32]
A Bayesian analysis of the Gumbel distribution - ResearchGate
Aug 10, 2025 · We considered an uninformative prior distribution for the location parameter, μ, and three different prior distributions for the scale parameter ...
[33]
[PDF] Categorical Reparameterization with Gumbel-Softmax
We begin by defining the Gumbel-Softmax distribution, a continuous distribution over the simplex that can approximate samples from a categorical distribution.
[34]
A Review of the Gumbel-max Trick and its Extensions for Discrete ...
Oct 4, 2021 · The Gumbel-max trick is a method to draw a sample from a categorical distribution, given by its unnormalized (log-)probabilities.Missing: class | Show results with:class
[35]
[PDF] A Review of the Gumbel-max Trick and its Extensions for Discrete ...
Feb 1, 2023 · Abstract—The Gumbel-max trick is a method to draw a sample from a categorical distribution, given by its unnormalized (log-) probabilities. Over ...
[36]
[PDF] Comparison of Prediction Intervals for the Gumbel Distribution
The problem of obtaining a prediction interval at specified confidence level to contain k future observations from the Gumbel distribution, ...Missing: finance | Show results with:finance
[37]
Managing extreme risks in tranquil and volatile markets using ...
In this paper, we apply both unconditional and conditional EVT models to the management of extreme market risks in stock markets.
[38]
[PDF] An Application of Extreme Value Theory for Measuring Financial Risk
The exponentially decaying tails of the Gumbel distribution characterize thin tailed distributions. Finally, the Weibull distribution is the asymptotic ...
[39]
[PDF] evd: Functions for Extreme Value Distributions - CRAN
Description Extends simulation, distribution, quantile and density functions to univariate and multivariate parametric extreme value distributions, and provides ...<|separator|>
[40]
scipy.stats.gumbel_r — SciPy v1.16.2 Manual
A right-skewed Gumbel continuous random variable. As an instance of the rv_continuous class, gumbel_r object inherits from it a collection of generic methods.
[41]
A discrete analog of Gumbel distribution: properties, parameter ...
A discrete version of the Gumbel distribution (Type-I Extreme Value distribution) has been derived by using the general approach of discretization of a ...
[42]
[PDF] Estimating Parameters of Gumbel Distribution using the Methods of ...
We derive here estimators for the parameters of the Gumbel distribution using three estimating methods, namely, the probability weighted moments, the moment and.
[43]
A review of methods of parameter estimation for the extreme value ...
Four important methods of estimating the parameters of the extreme value type-1 (Gumbel) distribution, namely: (1) moments (MMM); (2) maximum likelihood (MML); ...
[44]
Monte Carlo Comparison of the Parameter Estimation Methods for ...
Abstract. The performances of the seven different parameter estimation methods for the Gumbel distribution are compared with numerical simulations.
[45]
mlgumbel Gumbel distribution maximum likelihood estimation
mlgumbel: Gumbel distribution maximum likelihood estimation. Description. Uses Newton-Raphson to estimate the parameters of the Gumbel distribution.<|control11|><|separator|>
[46]
Comparison of different estimation methods for extreme value ... - NIH
The main objective of this paper is to determine the best estimators of the unknown parameters of the extreme value distribution.
[47]
Implement the Gumbel distribution in SAS - The DO Loop
Jul 24, 2019 · QUANTILE: F-1(p; μ, σ) = μ - σ log(-log(p)), where LOG is the natural logarithm. The RAND function in Base SAS and the RANDGEN function in SAS/ ...
[48]
[PDF] Non- Uni form - Random Variate Generation
The plot evolves around the expected complexlty of random variate genera- tlon algorlthms. We set up an ldeallzed computatlonal model (without overdolng lt), ...
[49]
[PDF] 1 Acceptance-Rejection Method
The acceptance-rejection method finds an alternative distribution G, close to the target distribution F, and uses an algorithm to generate from G. The ratio f( ...<|separator|>
[50]
[PDF] A Short History of Markov Chain Monte Carlo
Sep 30, 2009 · Monte Carlo methods were born in Los Alamos, New Mexico during World War II, eventually resulting in the Metropolis algorithm in the early 1950s ...
[51]
Probability weighted moments compared with some traditional ...
Results were derived from Monte Carlo experiments by using both independent and serially correlated Gumbel numbers.
[52]
Categorical Reparameterization with Gumbel-Softmax - arXiv
Nov 3, 2016 · We show that our Gumbel-Softmax estimator outperforms state-of-the-art gradient estimators on structured output prediction and unsupervised ...