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Questions tagged [gumbel]

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

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Joint probability distribution of some Gumbel differences

I would like your help to double check my derivations below involving the joint probability distribution of some Gumbel differences. Consider $K$ i.i.d. random variables $\epsilon_1,...,\epsilon_K$, ...
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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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1answer
42 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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84 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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250 views

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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124 views

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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428 views

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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72 views

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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221 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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242 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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596 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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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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119 views

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
351 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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1answer
135 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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1answer
463 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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1answer
254 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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2answers
2k views

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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1answer
186 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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782 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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809 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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256 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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1answer
12k views

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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617 views

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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708 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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15k views

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^{...