Questions tagged [gumbel]

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

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CDF for the linear combination of p Gumbel random variables

I'd like to know if there is an article or any insights to derive the exact CDF for the linear combination of p Gumbel random variables, as it was shown for the PDF in this article. Let $Z = \...
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56 views

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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Generating function of relaxed Bernoulli

In [1,2], an approximate continuous relaxation of the Bernoulli distribution is introduced as follows: $$X = \frac{1}{1 + e^{- (\theta - L) / \tau}}$$ where $L$ is a Logistic random variable, $\tau&...
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56 views

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 ...
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95 views

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[-\...
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1answer
48 views

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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74 views

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, ...
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143 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 ...
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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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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}...
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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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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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254 views

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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305 views

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 $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$. ...
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764 views

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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Extreme value distribution with unknown variance

Let $\{X_1,\ldots,X_n\}$ be a sequence of r.v. such that $X_i\sim N(0,\sigma^2)$. It is usually stated in Extreme Value Theory textbooks that (for suitably chosen $a_n$ and $b_n$) $$\mathbb{P}\left(\...
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185 views

t-test with logistic and Gumbel distributions

I know that one of the basic assumptions of a t-test is that the data is drawn from a Gaussian distribution. Using an Anderson-Darling test, I've found that the datasets I am working with are either ...
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CDF for Correlated Gumbel Distribution in Nested Logit simulation

I am interested in estimating a nested logit model following McFadden (1978)'s formulation. It is simple to numerically verify the result in a standard situation with independently drawn error terms, ...
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How to implement MLE of Gumbel Distribution

I'm trying implement the Maximum Likelihood Estimation in R to Gumbel distribution, but the algorithm doesn't converge. I'm using this parametrization to gumbel: $${\frac {1}{\sigma}{{\rm e}^{{\...
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Which PDF of X leads to a Gumbel distribution of the finite-size average of X?

Consider the statistic "average of $N$ idd random variables $X_i$", $$S_N = \frac{1}{N} \sum_{i=1}^N X_i$$ Consider also that, by a numerical experiment, it is observed that the distribution of $...
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Weibull, Gumbell and Extreme Value: from mean and variance to shape, scale and location parameter

I need to sample random numbers from Weibull, Gumbel and Generalized extreme value distributions. Of all of these distributions I know mean and variance. My question is: how can I determine these ...
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1answer
459 views

Pairwise comparison with Bradley-Terry

I am performing a pairwise comparison test for the perceived weight of objects. I want to estimate the difference between each pair, say, A - B. I suspect that the underlying distributions of A, and B,...
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Expectation of the Maximum of iid Gumbel Variables

I keep reading in economics journals about a particular result used in random utility models. One version of the result is: if $\epsilon_i \sim_{iid}, $ Gumbel($\mu, 1), \forall i$, then: $$E[\max_i(...
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183 views

Generate tail of distribution by a given sample in R

I have a sample of measurements from a real life device which misses all the measurements that are less than some threshold (given device is not precise enough). From theory and also measurements ...
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554 views

The distribution of the maximum of N independent but not identically distributed Gumbel random variables

I am interesting in determining if there is a closed form expression of the CDF and PDF of the maximum on $N$ Gumbel distributions that are independent but not identically distributed. In particular, ...
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277 views

How to derive formula for marginal probability of choosing nest in nested logit model?

I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n ...
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Maximum of Independent Gamma random variables? [closed]

Suppose $Y=\max\{X_1, X_2,\dots,X_N\}$ where all $X_i$ are independent and follows gamma distribution. I know that extreme value theory deals with maximum of random variables. Can anybody tell me, ...
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Distribution with a given moment generating function

As a follow-up to a question on a central limit theorem for independent random variables (r.v.) here, let $Y_j=-\log(1-V_j)$, where $V_j\sim\mbox{beta}(1-\sigma,j\sigma)$, $j\in\mathbb{N}^*$, $\sigma\...
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Relationship between Gumbel and Weibull distribution, accelerated failure time models, and Survreg using R

I have three questions concerning accelerated failure time models (AFT), one statistical, one regarding how to implement these models in R, and one related to finding out information about what R is ...
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191 views

Can somebody identify this distribution?

I am searching for the name of the distribution associated with this density on $\mathbb{R}_+$: $$p(r|\lambda) = \frac{2\lambda r\exp\left(\lambda\exp\left(-r^{2}\right)-r^{2}\right)}{\exp\left(\...
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813 views

Gumbel Copula generation using nonparametric correlations like Kendall's tau

I have 2 different variates W,X. I want to compute Gumbel copula for these variates. I followed following steps for the same: 1. To compute kendall's tau I used R's package Kendall. From kendall's tau ...
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928 views

Copula generation (Gaussian, t and Gumbel) with the help of correlation matrix using R

I have a set of data of 2 variates. I have generated correlation matrix between the variates. Using copula package of R, I computed t-copula using correlation ...
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294 views

Normalising Constant for exponentiated function

What would the normalising constant be of the following, or atleast an approximation? I would like to avoid sampling. $$f(\theta)=\exp(-k_1e^{-k_2\theta^2}-\theta^2)\qquad\theta\in(-\infty,\infty), \...
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Usable estimators for parameters in Gumbel distribution

The Gumbel distribution has the general form: $$F(y)=\exp\left({-\exp{\left(-\frac{y-\mu}{\sigma}\right)}}\right), \quad y\in \mathbb{R}$$ where $\mu \in \mathbb{R}$ and $\sigma >0$. Let $W_1,...,...
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Confidence intervals for extreme value distributions

I have wind data that i'm using to perform extreme value analysis (calculate return levels). I'm using R with packages 'evd', 'extRemes' and 'ismev'. I'm fitting GEV, Gumbel and Weibull distributions,...
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Is there a conjugate prior for Gumbel-distributed data?

I am trying to fit a Gumbel distribution to a set of N i.i.d. data points. I was wondering whether there is a conjugate prior to infer the two parameters of the Gumbel pdf.
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735 views

How to identify outliers from a Gumbel distribution with known parameters?

I have a model for data in my experiment that states that the data has a Gumbel distribution with known location and scale. I am then looking at observations with very high scores that I suspect to ...
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EM maximum likelihood estimation for Weibull distribution

Note: I am posting a question from a former student of mine unable to post on his own for technical reasons. Given an iid sample $x_1,\ldots,x_n$ from a Weibull distribution with pdf $$ f_k(x) = k x^{...