Questions tagged [gumbel-distribution]
The Gumbel distribution (or generalized extreme-value distribution type I) is used in modeling of extrema.
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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 ...
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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 ...
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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 ...
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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 ...
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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]/\...
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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 ...
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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[-\...
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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, ...
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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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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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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. $\epsilon_{1}, \epsilon_{0} \overset{\text{iid}}{\sim} \text{Gumbel}(\mu,\beta)$:
$$F(\epsilon) = \exp(-\exp(-\frac{...
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Extreme value distribution with unknown variance
Let $\{X_1,\ldots,X_n\}$ be a sequence of r.v. such that $X_i\sim N(0,\sigma^2)$.
It is usually stated in Extreme Value Theory textbooks that (for suitably chosen $a_n$ and $b_n$)
$$\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:
$${\frac {1}{\sigma}{{\rm e}^{{\...
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Which PDF of X leads to a Gumbel distribution of the finite-size average of X?
Consider the statistic "average of $N$ idd random variables $X_i$",
$$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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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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