Questions tagged [gumbel-distribution]
The Gumbel distribution (or generalized extreme-value distribution type I) is used in modeling of extrema.
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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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1answer
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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 ...
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Scheduling during training of Gumble Softmax
Is there a best practice for scheduling the temperature decrease in Gumble softmax? Setting the temperature too low too quickly leads to high variance in the gradients and unstable training. My ...
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Usable estimators for parameters in Gumbel (minimum) distribution
As eloquently outlined here, the usable estimators have been estimated for the gumbel maximum distribution. My question is how you would do the same for the gumbel minimum distribution?
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Conditional probability that one Gumbel random variable is greater than another
Suppose I have three random variables drawn independently from Gumbel distributions (Wikipedia) with different means but the same scale parameter:
$$
X_1 \sim \text{Gumbel}(\mu_1, \beta) \\
X_2 \sim \...
2
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1answer
84 views
Standard error with maximum likelihood estimation
ASTM E2238 describes the prediction of the largest inclusion in a polished steel surface of 150,000 mm² based on a Gumbel distribution of the largest inclusions measured in each of 24 polished ...
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negative values for AIC and BIC
I am trying to fit a gumbel distribution using MLE for the following 10 data points.
DATA=(3.62,3.76,3.57,3.56,3.61,3.77,3.46,3.6,3.39,3.74)
The problem is that the ...
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problem with PyMC3 for evaluation of parameters of gumbel distribution
In the following code, i am trying to evaluate the parameters of a gumbel distribution using PyMC3.
The data is annual_maxima_1 and the method works for MLE but i could not make any conclusions ...
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Derive the closed form of a probability expression
Consider the following probability
$$
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 )
$$
where
$\lambda\...
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votes
1answer
103 views
Why do we need Gumbel distribution?
I check the Gumbel distribution article on Wikipedia, it says it is useful to represent the distribution of maxima. But it is not easy to understand how it works? A detailed explanation or examples ...
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2answers
129 views
Fourth class of extreme-value distributions?
The generalized extreme-value distribution encompasses three classes of distributions:
Frechet, which are regularly varying, infinite right limit.
Gumbel, which are not regularly varying, infinite ...
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556 views
Extreme Value (Gumbel) distribution a member of exponential family
This is a question for discussion in my Linear Model class. I am having a hard time showing that the distribution belongs to the exponential family
PDF: $f(y; \theta) = 1/\varphi \exp([y − \theta]/\...
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134 views
KL divergence for Generalized Extreme Value distribution
I have found a derivation for the Kullback–Leibler divergence between 2 Gumbel distributions here:
http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf on page 64
That document also has a ...
2
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1answer
136 views
Convergence maximum of Normal rv to gumbel through simulation (metropolis hastings)
I would like to see the convergence of an order statistic to its respective Extreme Value attractor by simulating with the Metropolis Hastings algorithm (I am self-studying MCMC algos).
I was trying ...
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1answer
101 views
Identification of discrete choice models
Consider the classical Logit model. In particular, let $\mathcal{Y}\equiv (0,1,...,L)$ be the set of options available to consumers, where $0$ denotes the outside option. Let
$$
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
86 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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1answer
100 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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1answer
218 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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211 views
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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1answer
388 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)...
7
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1answer
2k 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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0answers
86 views
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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1answer
126 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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1answer
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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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1answer
356 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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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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2answers
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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 $b_n$)
$$\mathbb{P}\left(\...
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2answers
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t-test with logistic and Gumbel distributions
I know that one of the basic assumptions of a t-test is that the data is drawn from a Gaussian distribution.
Using an Anderson-Darling test, I've found that the datasets I am working with are either ...
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CDF for Correlated Gumbel Distribution in Nested Logit simulation
I am interested in estimating a nested logit model following McFadden (1978)'s formulation. It is simple to numerically verify the result in a standard situation with independently drawn error terms, ...
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2answers
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How to implement MLE of Gumbel Distribution
I'm trying implement the Maximum Likelihood Estimation in R to Gumbel distribution, but the algorithm doesn't converge.
I'm using this parametrization to gumbel:
$${\frac {1}{\sigma}{{\rm e}^{{\...
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votes
2answers
255 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 ...
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0answers
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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
550 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
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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
719 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
328 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
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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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2answers
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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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1answer
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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
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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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1answer
905 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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1answer
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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 ...