Questions tagged [gumbel-distribution]
The Gumbel distribution (or generalized extreme-value distribution type I) is used in modeling of extrema.
52
questions
0
votes
0answers
31 views
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 ...
0
votes
0answers
9 views
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 ...
3
votes
0answers
37 views
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\...
6
votes
1answer
80 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 ...
1
vote
2answers
111 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 ...
0
votes
0answers
10 views
Chains did not converge and mix well
I am trying to compute the parameter of copula model (Gumbel copula). I have used zero/one trick to write the copula likelihood model. The chains did not converge or mix at all. I have tried several ...
0
votes
0answers
279 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]/\...
0
votes
0answers
51 views
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 = \...
0
votes
0answers
95 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
votes
1answer
88 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 ...
1
vote
1answer
99 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{...
3
votes
0answers
287 views
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 ...
1
vote
0answers
12 views
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[-\...
2
votes
1answer
63 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 ...
1
vote
0answers
25 views
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 ...
0
votes
1answer
90 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 ...
1
vote
0answers
245 views
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, ...
1
vote
1answer
179 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 ...
1
vote
0answers
183 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_\...
5
votes
1answer
1k 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}...
6
votes
1answer
265 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)...
5
votes
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($\log\alpha_j+g_j$) ...
2
votes
0answers
84 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 ...
0
votes
1answer
110 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:
...
3
votes
1answer
266 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 \...
1
vote
0answers
106 views
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 \...
4
votes
0answers
121 views
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-...
1
vote
1answer
329 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 $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$. ...
9
votes
1answer
878 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. $\epsilon_{1}, \epsilon_{0} \overset{\text{iid}}{\sim} \text{Gumbel}(\mu,\beta)$:
$$F(\epsilon) = \exp(-\exp(-\frac{...
3
votes
2answers
95 views
Extreme value distribution with unknown variance
Let $\{X_1,\ldots,X_n\}$ be a sequence of r.v. such that $X_i\sim N(0,\sigma^2)$.
It is usually stated in Extreme Value Theory textbooks that (for suitably chosen $a_n$ and $b_n$)
$$\mathbb{P}\left(\...
1
vote
2answers
209 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 ...
2
votes
0answers
162 views
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, ...
4
votes
2answers
3k views
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:
$${\frac {1}{\sigma}{{\rm e}^{{\...
3
votes
2answers
215 views
Which PDF of X leads to a Gumbel distribution of the finite-size average of X?
Consider the statistic "average of $N$ idd random variables $X_i$",
$$S_N = \frac{1}{N} \sum_{i=1}^N X_i
$$
Consider also that, by a numerical experiment, it is observed that the ...
1
vote
0answers
244 views
Weibull, Gumbell and Extreme Value: from mean and variance to shape, scale and location parameter
I need to sample random numbers from Weibull, Gumbel and Generalized extreme value distributions. Of all of these distributions I know mean and variance. My question is: how can I determine these ...
3
votes
1answer
509 views
Pairwise comparison with Bradley-Terry
I am performing a pairwise comparison test for the perceived weight of objects. I want to estimate the difference between each pair, say, A - B. I suspect that the underlying distributions of A, and B,...
15
votes
2answers
4k views
Expectation of the Maximum of iid Gumbel Variables
I keep reading in economics journals about a particular result used in random utility models. One version of the result is: if $\epsilon_i \sim_{iid}, $ Gumbel($\mu, 1), \forall i$, then:
$$E[\max_i(...
6
votes
1answer
231 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 ...
1
vote
1answer
660 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, ...
0
votes
1answer
298 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 ...
6
votes
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, ...
8
votes
0answers
331 views
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\...
7
votes
0answers
1k views
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 ...
7
votes
1answer
197 views
Can somebody identify this distribution?
I am searching for the name of the distribution associated with this density on $\mathbb{R}_+$:
$$p(r|\lambda) = \frac{2\lambda r\exp\left(\lambda\exp\left(-r^{2}\right)-r^{2}\right)}{\exp\left(\...
1
vote
1answer
862 views
Gumbel Copula generation using nonparametric correlations like Kendall's tau
I have 2 different variates W,X. I want to compute Gumbel copula for these variates. I followed following steps for the same:
1. To compute kendall's tau I used R's package Kendall. From kendall's tau ...
0
votes
1answer
1k views
Copula generation (Gaussian, t and Gumbel) with the help of correlation matrix using R
I have a set of data of 2 variates. I have generated correlation matrix between the variates. Using copula package of R, I computed t-copula using correlation ...
0
votes
0answers
310 views
Normalising Constant for exponentiated function
What would the normalising constant be of the following, or atleast an approximation? I would like to avoid sampling.
$$f(\theta)=\exp(-k_1e^{-k_2\theta^2}-\theta^2)\qquad\theta\in(-\infty,\infty), \...
5
votes
1answer
13k 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,...,...
3
votes
0answers
1k views
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,...
4
votes
0answers
770 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.
