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Questions tagged [gumbel-distribution]

The Gumbel distribution (or generalized extreme-value distribution type I) is used in modeling of extrema.

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Distribution of a random variable conditional on its being a maximum or not

Consider the random variables $\epsilon_1,\dots, \epsilon_D$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Assume they are continuous. Let $$ Y=\sum_{d=1}^D d\times \mathbb{1}\{\...
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Multinomial Logit Extension

The derivation of the multinomial logit probabilities depends on the difference of two Type 1 extreme value (Gumbel) random variables following a logistic distribution. We say the unobserved utility ...
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Validity of bootstrapping for estimation of annual maxima distribution

I am working with a large timeseries (millions data points) spread across 5 years from which I would like to estimate the annual maxima distribution and subsequently a quantile of this distribution. ...
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Extreme value function in R

I have a small data set (n=25) with an unknown distribution and I'm trying to find a distribution function that is as similar as possible in order to ultimately determine tolerance limits. I used the ...
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Why use Gumbel softmax instead of taking the (soft)argmax of the logits (or softmax then argmax)?

My understanding is that the goal of using Gumbel softmax is to change an output that contains logits into a one-hot vector corresponding to the highest probability choice (based on those logits). ...
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Empirically estimating extremal coefficient using minima of Fréchet margins

I recently came across a paper which uses the following formula to empirically estimate the extremal correlation coefficient $\chi_{ij}$ between two variables $x$ and $y$ as follows: $$ \chi_{xy} = \...
ThreeOrangeOneRed's user avatar
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Why does the best fitting Weibull distribution for this survival data deviate further from the actual data than a poorly fit distribution?

I started working with the Gumbel distribution and fit it to the lung dataset to try it out. I then compared it with the survival curve using the Weibull ...
Village.Idyot's user avatar
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Is there a clean way to derive the start parameters for running the `fitdist()` function for the Gumbel distribution?

I've begun working with the Gumbel distribution and started with the example in the package documentation at https://cran.r-project.org/web/packages/fitdistrplus/fitdistrplus.pdf. For the Gumbel ...
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How to calculate Gumbel with LMoments and GEV with method of moments

I need to calculate the values for certain return periods of a flood event (up to 5000). It has to be GEV with method of moments and Gumbel with L-Moments. But I am not sure about how to calculate ...
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Negative Log inside Negative Log

Is there a name for a function which is like $f(x)=-\log(-\log(x))$ where $0<x<1$? Or, is there any name for this function $g(x)=x+\log(\frac{1}{x})$ where $0<x$? Exchanging $x=-\log k$ in $g(...
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What does the parameter s in Sum-of-Gamma mean?

So the sum of gamma, introduced in I-MLE is defined as the following: $$SoG(k,t,s)=\frac{t}{k} \left( \sum^{s}_{i=1} Gamma(1/k,k/i) - \log(s) \right)$$ But what exactly is $s$? It clearly controls the ...
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Bivariate Gumbel Distribution - Joint Survival function

I need help for this exercise please: How do I compute the joint Survival function for the bivariate Gumbel distribution: $f_{xy} = ((1+\delta x)(1+\delta y)-\delta) e^{-x-y-\delta xy}$ According to ...
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Calculating confidence Interval for a return time curve, via non-parametric bootstrapping

I have some precipitation data (yearly extremes), which I have fit with a Gumbel distribution (CDF), from which I have calculated a return time distribution. I want to calculate the 95% confidence ...
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Compute joint probability of two variables without using copulas

I have two timeseries one for a three-day accumulated rainfall and another one for a daily storm surge height. I would like to calculate the probability of a certain rainfall depth with a certain ...
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Fitting Gumbel distribution based the maximal observation

Assume that we only consider $$G(x)=\exp(-\exp(\frac{x-\mu}{\sigma}))$$ is the Gumbel distribution. Question: Suppose we have a set of maximum values $\{Y_i\}_{i=1}^m$, why can the article directly (...
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How can I compare the MLE by standard error?

In the paper Extreme value theory based on the r largest annual events (page 32), the idea is that he wants to fit the Gumbel distribution using a dataset. In this dataset, we have the largest ten ...
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Can we fit extreme value distribution by build-in package?

I try to find a package in R to fit Gumbel distribution by Block Maxima Approach using maximal likelihood function (see here) $$ G(x; \mu , \sigma)=\exp[-e^{-\frac{x-\mu}{\sigma}}]. $$ The block ...
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Why does Gumbel distribution have two different expressions?

Let $X_1,X_2,\dots,X_n$ be iid random variables with distribution function $F(x)$ and $M_n:=\max\{X_1,\dots,X_n\}$. By the extreme value theorem, there exist two sequences of real numbers $a_n>0$ ...
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General Closed Form and dispersion parameter of the Expected Maximum of i.i.d Gumbel Variables

I would like to know the general closed form of the expected maximum of i.i.d Gumbel variables. I found this onlie: Expectation of the Maximum of iid Gumbel Variables In the linked page, it shows the ...
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Why Gumbel softmax and not other types of softmax?

Ignore the method name. Is there a reason why in the Gumbel softmax trick we sample from Gumbel distribution? Since we are doing something similar to a reparameterization trick, can't we just sample ...
Junhan Ouyang's user avatar
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Bayesian parameter estimation using priors from MLE-parameters

10 Points of maximum water levels available ONLY. I am trying to fit the data to a gumbel/GEV distribution using the following method. -MLE -Bayesian Problem When i try to fit the data and get the ...
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Derivation of moment generating function for limiting distribution of sum of logbeta distributed variables

A sum of logbeta distributed variables occurs in this question Distribution with a given moment generating function Let, $X_j \sim Beta(j\sigma, 1-\sigma)$, $Y_j = -\log(X_j)$ and $S_n = \sum_{j=1}^n ...
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Is there an intuition to the mean of a Gumbel distribution being the Euler constant vis-à-vis the modeling of extreme events?

The derivation is pure mechanistic integration as in here, and it doesn't come as a surprise to find the Euler constant in a distribution such as $\Lambda(x)=e^{-e^{-x}}$. However, the Euler constant ...
Antoni Parellada's user avatar
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Standard error with maximum likelihood estimation

ASTM E2238 describes the prediction of the largest inclusion in a polished steel surface of 150,000 mm² based on a Gumbel distribution of the largest inclusions measured in each of 24 polished ...
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negative values for AIC and BIC

I am trying to fit a gumbel distribution using MLE for the following 10 data points. DATA=(3.62,3.76,3.57,3.56,3.61,3.77,3.46,3.6,3.39,3.74) The problem is that the ...
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Derive the closed form of a probability expression

Consider the following probability $$ A\equiv Pr\Big (\delta_1+v+\lambda \epsilon_1\geq \delta_2+v+\lambda \epsilon_2 \text{, } \delta_1+v+\lambda \epsilon_1\geq \epsilon_0 \Big ) $$ where $\lambda\...
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Why do we need Gumbel distribution?

I check the Gumbel distribution article on Wikipedia, it says it is useful to represent the distribution of maxima. But it is not easy to understand how it works? A detailed explanation or examples ...
Qinsheng Zhang's user avatar
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221 views

Fourth class of extreme-value distributions?

The generalized extreme-value distribution encompasses three classes of distributions: Frechet, which are regularly varying, infinite right limit. Gumbel, which are not regularly varying, infinite ...
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Extreme Value (Gumbel) distribution a member of exponential family

This is a question for discussion in my Linear Model class. I am having a hard time showing that the distribution belongs to the exponential family PDF: $f(y; \theta) = 1/\varphi \exp([y − \theta]/\...
eugelio's user avatar
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KL divergence for Generalized Extreme Value distribution

I have found a derivation for the Kullback–Leibler divergence between 2 Gumbel distributions here: http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf on page 64 That document also has a ...
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Convergence maximum of Normal rv to gumbel through simulation (metropolis hastings)

I would like to see the convergence of an order statistic to its respective Extreme Value attractor by simulating with the Metropolis Hastings algorithm (I am self-studying MCMC algos). I was trying ...
Vittorio Apicella's user avatar
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Identification of discrete choice models

Consider the classical Logit model. In particular, let $\mathcal{Y}\equiv (0,1,...,L)$ be the set of options available to consumers, where $0$ denotes the outside option. Let $$ u_y\equiv \begin{...
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Convergence rate of the maximum of Weibull random variables to a Gumbel distribution

Given a sequence of iid samples $X_1, \dots, X_n,$ where each $X_i$ comes from a Weibull distribution with shape parameter $k$ and scale parameter $\lambda$. Then it is a well-known result that the ...
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Simulating Draws of Multivariate EV-Type Distribution

Let $\varepsilon = [\varepsilon_1,...,\varepsilon_J]$ be a random vector that we can partition into $K$ disjoint subvectors. $\varepsilon$ has this cdf: \begin{equation} F(\varepsilon) = \exp \bigg[-\...
Matt's user avatar
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When a function of two random variables is Gumbel?

Consider two random variables $X,Y$. Is there any example in which $X$ and $Y$ have a known parametric distribution such that $f(X,Y)$ is Gumbel with scale $\sigma$ and location $\beta$, for some ...
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fit a gumbel copula to 500 set generated random number

i have a question, of finding the tail coefficient of gumbel copula. I generated 500 set of random variable, with 4 different theta of 1, 1.5, 2 and 3. Then I fit them to gumbel copula with maximum ...
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Interpretation of a Gumbel distribution's results

I am using (essentially) the approach outlined in the paper "Statistical-based WCET estimation and validation" (http://drops.dagstuhl.de/opus/volltexte/2009/2291/pdf/Hansen.2291.pdf) to build a Gumbel ...
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Did I understand the usage of Gumbel-Softmax reparametrization correctly?

I am working on a deep learning model, which has a mixture of experts formulation like $\log p(y|x)=\log \sum_{z}p(y|z,x,\theta)p(z|x,\phi)$. So, each $p(y|z,x,\theta)$ is a deep learning classifier, ...
Ufuk Can Bicici's user avatar
1 vote
1 answer
347 views

How to derive joint CDF Gumbel distribution

If you have 3 random variables: $X$, $Y$, and $Z$ and they have independent Gumbel distribution. $A$, $B$ and $C$ are three discrete random variables that are functions of $X$, $Y$, and $Z$ as per the ...
Tres Roman's user avatar
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Why Monte Carlo sampling is not needed for reparameterization trick?

To esitimate $\nabla_\theta \mathbb{E}_{z\sim p_\theta(z)}[f(z)]$, we have two options: REINFORCE: $\nabla_\theta \mathbb{E}_{z\sim p_\theta(z)}[f(z)] = \mathbb{E}_{z\sim p_\theta(z)}[ f(z)\nabla_\...
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1 answer
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How to find the Inverse Transform of the Gumbel distribution

How does one find the Inverse Tranform of the Gumbel distribution? Let $X\sim \text{Gumbel}(\mu,\beta)$ with scale parameter $\beta>0$. The CDF is then $F_X(x)=\text{e}^{-\text{e}^{-(x-\mu)/\beta}...
SecretAgentMan's user avatar
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1 answer
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Extreme Value Theory - domains of attraction and techniques for evaluting a limit

We consider the gamma uniform G distribution as specified by Torabi and Montazeri: $$f(x) = \frac{1}{\Gamma (a)}\frac{g(x)}{[1-G(x)]^2}\left[\frac{G(x)}{1-G(x)}\right]^{a-1}\exp\left[\frac{G(x)}{1-G(x)...
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Why do we need the temperature in Gumbel-Softmax trick?

Assuming a discrete variable $z_j$ with unnormalized probability $\alpha_j$, one way to sample is to apply argmax(softmax($\alpha_j$)), another is to do the Gumbel trick argmax($\log\alpha_j+g_j$) ...
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Interpret the result of a fitted non-stationary Gumbel model

I have a dataset on wildfires that I fitted to a Gumbel distribution with a set of covariates (using the gevrFit function in the eva package in R). The result of ...
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1 answer
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Simulate >2 variables from Gumbel Copula

I'm trying to simulate multiple random variables with different taus from the Gumbel copula. For the normal copula it's pretty simple, eg: ...
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Extreme value theory: show that $ \lim_{n\rightarrow \infty}a_n $ exists and is finite

Well known facts in extreme value theory: Let $\{X_i\}_{\forall i \in \{1,...,n\}}$ be i.i.d. random variables with cdf $F$. If there exists $\{a_n\}_{n\in \mathbb{N}}>0$, and $\{b_n\}_{n\in \...
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Normalising constant of the Gumbel in extreme value theory

Well known facts in extreme value theory: Let $\{X_i\}_{\forall i \in \{1,...,n\}}$ be i.i.d. random variables with cdf $F$. If there exists $\{a_n\}_{n\in \mathbb{N}}>0$, and $\{b_n\}_{n\in \...
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Extreme value distribution for univariate normal: Derive parameters of the Gumbel [duplicate]

I have a question regarding the extreme value distribution corresponding to i.i.d. samples $X_i$ from a normal distribution, say $X_i\sim N(\mu, \sigma^2)$. According to the theorem of Fisher-Tippett-...
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1 answer
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Mean and Standard deviation of Gumbel distributions subtraction

I know that two normal distributions can be subtracted and get a new distribution with a mean of $\bar{x} = \bar{x}_1-\bar{x}_2$ and a standard deviation of $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$. ...
Chris's user avatar
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11 votes
2 answers
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Conditional expectation of a truncated RV derivation, gumbel distribution (logistic difference)

I have two random variables which are independent and identically distributed, i.e. $\epsilon_{1}, \epsilon_{0} \overset{\text{iid}}{\sim} \text{Gumbel}(\mu,\beta)$: $$F(\epsilon) = \exp(-\exp(-\frac{...
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