The Gumbel distribution (or generalized extreme-value distribution type I) is used in modeling of extrema.
Wikipedia has an article https://en.wikipedia.org/wiki/Gumbel_distribution with further references.
Questions tagged [gumbel-distribution]
57 questions
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votes
4 answers
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
$$
f_k(x) = k x^{k-1} e^{-x^k} \quad x>0
$$
is there a useful…
Xi'an
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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:
$$E[\max_i(\delta_i + \epsilon_i)] = \mu + \gamma +…
Jason
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Relationship between Gumbel and Weibull distribution, accelerated failure time models, and Survreg using R
I have three questions concerning accelerated failure time models (AFT), one statistical, one regarding how to implement these models in R, and one related to finding out information about what R is doing. In short my questions are;
1) What is the…
dandar
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Conditional expectation of a truncated RV derivation, gumbel distribution (logistic difference)
I have two random variables which are independent and identically distributed, i.e. $\epsilon_{1}, \epsilon_{0} \overset{\text{iid}}{\sim} \text{Gumbel}(\mu,\beta)$:
$$F(\epsilon) = \exp(-\exp(-\frac{\epsilon-\mu}{\beta})),$$
$$f(\epsilon) =…
wolfsatthedoor
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9
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2 answers
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 $V_j\sim\mbox{beta}(1-\sigma,j\sigma)$, $j\in\mathbb{N}^*$, $\sigma\in(0,1)$.
The shifted sums…
julyan
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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($\log\alpha_j+g_j$) where $g_j$ is gumbel generated noise. This second…
user3639557
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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 $W_1,...,W_n$ be random variables with that distribution and…
Raxel
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7
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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:
$$f(x) = \frac{1}{\Gamma (a)}\frac{g(x)}{[1-G(x)]^2}\left[\frac{G(x)}{1-G(x)}\right]^{a-1}\exp\left[\frac{G(x)}{1-G(x)}\right]$$
where $G$ is a valid c.d.f., $g$ the…
Will
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7
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2 answers
Maximum of Independent Gamma random variables?
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, hopefully with reference, $Y$ will follow which…
upol94
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Can somebody identify this distribution?
I am searching for the name of the distribution associated with this density on $\mathbb{R}_+$:
$$p(r|\lambda) = \frac{2\lambda r\exp\left(\lambda\exp\left(-r^{2}\right)-r^{2}\right)}{\exp\left(\lambda\right)-1}.$$
It arises from the mixture…
fabee
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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 appears in many places and is a reference in…
Antoni Parellada
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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 may be more helpful.
I notice the pdf of the…
Qinsheng Zhang
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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 from more precise devices I know that the data…
Igor Korotkov
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1 answer
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 $F_X(x)=\text{e}^{-\text{e}^{-(x-\mu)/\beta}}$.
SecretAgentMan
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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 $\{b_n\}_{n\in \mathbb{N}}\in \mathbb{R}$ such that $Z_n\equiv…
TEX
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