All Questions
Tagged with extreme-value extreme-value or
606 questions
4
votes
2
answers
1k
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Thin tails and the Generalized Pareto
Rewriting my question:
On this Mathworks page:
https://www.mathworks.com/help/stats/generalized-pareto-distribution.html
it is said (as many textbooks say) that
"Distributions whose tails decrease ...
- 533
2
votes
1
answer
395
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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 ...
2
votes
1
answer
580
views
Distribution with 3 Modes, Find the 2 In-Between Minima
Suppose I have a dataset consisting of numbers drawn from three normal distributions $\mathcal N\!(\mu_{\rm left}, \sigma_{\rm left}^2),\ \mathcal N\!(\mu_{\rm center}, \sigma_{\rm center}^2),\ \...
- 59
3
votes
0
answers
555
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 ...
- 93
1
vote
1
answer
95
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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[-\...
- 11
2
votes
1
answer
83
views
Detecting extrema under uncertainty?
An old version of this question was poorly articulated. Here is another go:
I have fifty objects. With a different, independent, unbiased scale for each object, I measure their weights 100 times each ...
- 21
2
votes
1
answer
1k
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How to calculate the cumulative distribution function of a GEV distribution when $1+\xi(x-\mu)/\sigma\le0$?
I don't have a stats background let alone one in extreme value theory, and I have what I imagine is a simple question but one I that haven't been able to find the answer to. The cumulative ...
- 21
1
vote
1
answer
133
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MAP versus Component-Wise Maximum Marginal
Suppose I have the joint distribution:
\begin{align}
p(\mathbf{x}) = p(x_1, ... , x_n)
\end{align}
The maximum a posteriori (MAP) solution is given by:
\begin{align}
\mathbf{x}_{MAP} = \arg \max p(\...
- 555
3
votes
1
answer
61
views
Problem with two correlated random normals
Imagine you have a two-dimensional multivariate normal random variable with $\mu = [0, 0]$ and $\Sigma\ = \begin{bmatrix}1 & r\\r & 1\end{bmatrix}$. (Conceptually, you have two random normal ...
- 131
2
votes
0
answers
55
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Extreme Value Theory - Determining the positive normalising constant in the Extremal Types Theorem
I am working through the following question and cannot seem to work out how the final result is obtained from the last inequality involving $a_n$. Can someone shed some light?
- 309
0
votes
0
answers
73
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How to minimise and expectation with respect to a parameter?
Suppose that $X$ is a random variable with distribution $G$. Let $H(X;\theta)$ be a parametric function with $\theta \in \Theta \subset {\mathbb R}^p$. I want to maximize the function
$$\varphi(\...
- 1
7
votes
2
answers
286
views
What is the distribution of a bivariate normal component conditional on the max of the other component?
Let $n$ be a large integer, and consider two independent multivariate Gaussian $n$-vectors $x, z$ with $x\sim\mathcal{N}\left(0,I\right),$ and $z\sim\mathcal{N}\left(0,\sigma^2 I\right)$. Let $y=x+z$. ...
- 563
1
vote
1
answer
52
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Two datasets with same length give different number of extremes
I have two datasets of a given variable x that have the same length, let's say 14600 values in total each one.
I need to extract the extreme observations within ...
- 344
2
votes
1
answer
451
views
Show that $nX_{(1)}$ is not consistent
Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency ...
- 1,397
3
votes
1
answer
4k
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Normalizations: dividing by maximum [closed]
I'd like to know what are the reasons and benefits of dividing all the values of a dataset by the maximum of the dataset. Are they referred by authors? This normalization is well known in gene ...
- 141
3
votes
1
answer
249
views
Limiting Distribution of $n\left[Y_n\right]$ where $Y_n$ is the minimum of a sample of size n from Uniform$\left(0,\theta\right)$ distribution
Suppose $X_1,X_2,\dots,X_n$ is a random sample from Uniform$(0,\theta)$ for some unknown $\theta > 0$. Let $Y_n$ be the minimum of $X_1,X_2,\dots,X_n$.
(a) Suppose $F_n$ is the CDF of $nY_n$. Show ...
0
votes
1
answer
104
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Identify dependent or independent blocks of time series (clusters)
maybe I am lost in translation but I need your help.
Description: Having long time series of two variables I create some blocks (or clusters) with the method of peak over threshold but I need to ...
- 1
1
vote
0
answers
117
views
Group comparison for extreme value data: which method is suitable?
I have measured Gaussian curvature data of 3D objects from two different groups, A and B. I would like to find out whether the objects differ in curvature.
The distribution of data values for each ...
- 75
1
vote
1
answer
176
views
Meaning of Extreme Value distribution vs. lowest/highest Order Statistic
How exactly does the meaning of the Extreme Value Distribution differ from the distribution of the lowest/highest (extreme) order statistics?
I understand that the extreme value distribution (EVD) ...
- 111
1
vote
1
answer
151
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Question regarding Extreme Value Theory and finding the distribution of X(n)
Hello stats stack exchange, I have a question regarding Order Statistics and the asymptotic distribution of $X_n$ which is the rv for max($X_1$, $X_2$,...,$X_n$) where $X_i$ are from some distribution....
- 41
0
votes
0
answers
59
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Generalized logistic distribution
I saw on wikipedia https://en.wikipedia.org/wiki/Generalized_logistic_distribution that when $\alpha<\beta$, generalized type IV logistic distribution can be written as:
$\frac{\exp(-\alpha x)}{(\...
- 565
7
votes
1
answer
243
views
Is there a random variable $X$ with positive support such that the ratio of the two smallest realizations of an iid sample goes to one?
Imagine I have given a random variable $X$ with supp$(X)=(0,\infty)$ and $\mathbb P(X \in (0,a))>0$ for any fixed $a>0$
Now given an iid sample $X_1,...,X_n$ - is it possible that
$$X^{(2)}/...
user244467
2
votes
2
answers
2k
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The relationship between GEV and GPD
source: Embrechts pg 165, 354
(3.52) G is Generalized Pareto Distribution
Base on that theorem, could I conclude that
1) if I have a data and the excess fit with Generalized Pareto Distribution, then ...
- 29
0
votes
0
answers
51
views
Estimate true mean of the maximum of N sample means
Let's say we have N distributions $\mathcal N(\mu_i, \sigma_i)$, each with unknown mean $\mu_i$ and unknown standard deviation $\sigma_i$, $i=0,...,N-1$.
For each $i$, $M$ independent random samples ...
- 1
0
votes
0
answers
552
views
When I fit my data with GEV, I got positive parameter, but when I fit it with GPD, I got negative parameter?
My data is the total annual precipitation in Australia. My purpose is to observe the extreme precipitation on the right end tail. When I fit my data with Generalized Extreme Value, I got positive ...
- 29
2
votes
0
answers
300
views
Find the limiting distribution of $(bn)^{-\frac{1}{\alpha}} X_{(n)}$
let $\{X_n\}_{n\geq 1}$ be a sequence of i.i.d random variables with common distribution $F$, and write
$X_{(n)}=\max\{X_1,\cdots , X_n\}$ , $n=1,2,\cdots$
(a) for $\alpha >0$ , $\lim_{x\rightarrow ...
- 1,349
6
votes
2
answers
2k
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What is element-wise max pooling?
I came across this term in the VoxelNet paper in relation to point cloud based object detection using machine learning. It is mentioned in figures 2&3 and in 2.2.1
I am familiar with 2d max ...
- 63
2
votes
1
answer
20
views
What is the variance of the least of several series?
Given a number of series (or images) that are independent and have a Poisson distribution with the same mean, what is the variance of the series generated from taking the point-by-point minimum?
i.e. ...
- 21
8
votes
1
answer
272
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distribution for scaled Maximum of n independent Weibulls for $n \to \infty$
Assume that $X_1, X_2,...\sim Weibull(\lambda, k) \quad iid.$, i.e.
$F(X_1\leq x) = 1-e^{-(\lambda x)^k}$
define $M_n:= \max\{X_1, ..., X_n\}$ and $\tilde{M}_n:=\frac{M_n-b_n}{a_n}$
according to ...
- 3,114
0
votes
0
answers
218
views
How to assess uncertainty in a Bayesian analysis?
I'm building a model to estimate recurrence of flood magnitudes via fitting a GEV distribution to flow data. The aim of the study is to compare stationary and non stationary models using a Bayesian ...
1
vote
0
answers
196
views
How is the minimum logarithmic loss calculated when initializing the XGBoost algorithm?
Suppose there are $5$ sample units, $2$ of which carry the feature $y=1$ to be predicted and three of which carry the feature $y=0$. So, $2$ are positive.
The XGBoost algorithm initializes with
$\...
- 11
-1
votes
1
answer
210
views
Estimation of an exponential parameter
I´m trying to figure out the pdf $f_\min(X_i)$ of $\min(X_i)$, where the distribution of the sample $X_1,...,X_n$ is $\mathcal{E}xp(\lambda)$, where $\lambda$ is the unknown parameter.
I tried with ...
- 1
4
votes
0
answers
223
views
How to prove that the prior for which Bayes rule is also the minimax rule, is the least favorable prior?
I have read in the book Mathematical Statistics: A Decision Theoretic Approach by Thomas Ferguson that The prior for which the Bayes rule is also minimax rule, then that prior is Least favorable prior....
- 51
2
votes
1
answer
180
views
Maximum A Posteriori Estimate
The formula for calculating the MAP estimate of a particular parameter, $p$, is given by the following: $p^{MAP} =$ argmax $P(p)P(p|x)$.
Now I am trying to do a question where I am told the prior ...
- 571
4
votes
1
answer
10k
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cross entropy loss max value
The cross entropy loss function for multiclass can be computed as:
$$-\sum\limits_{i=1}^N y_i log \hat{y}_i$$
where $y_i$ is a class and $\hat{y}_i$ the estimated probability. The minimum value is $0$ ...
- 747
2
votes
0
answers
62
views
Normal equations issue
Let $A\mathbf{x}=\mathbf{b}$ be an overdetermined system, with $A$ being an $n \times m $ full-column rank rectangular matrix.
Are these minimization problems equivalent?
$$ 1) \;\underset{\mathbf{x}...
- 437
3
votes
1
answer
5k
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Expectation of max of two normal random variables
I have been reading this paper about the maximum and minimum of two normal distributed variables.
Inside the paper there is the formula for the expectation of this the maximum of the two variables. ...
- 133
1
vote
1
answer
41
views
How to compare frequencies of categorical variable with 3 possible values
There is one variable which can get one of 3 values and one sample. Lets assume values are A, B, C and frequencies are x, y, and z. How could I find if x > max(y,z), statistically significant? Or, in ...
- 11
5
votes
1
answer
122
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Identity on expectation of the minimum of two iid random variables with bounded support
I am reading the 2008 annals of statistics paper "Ranking and empirical minimisation of U-statistics" by Clémençon et. al, and read a statement which I do not know why is true. In order to accurately ...
- 657
4
votes
1
answer
1k
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m out of n bootstrap implementation in R
I am wishing to estimate the sampling distribution of an extreme order statistic (the sample maximum). The usual nonparametric (n-out-of-n) bootstrap fails miserably in this case.
Chernick (2011) ...
- 1,649
2
votes
2
answers
1k
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Estimating the min and max of a distribution
I have a measurement problem where I am attempting to measure the minimum and maximum height of a surface by taking point samples of heights. If I then look at the distribution of all height values, ...
- 177
0
votes
1
answer
159
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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 ...
- 131
0
votes
0
answers
24
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Variability metric for Top 3 sports teams in a league
**I am a bit unsure what to mark this as/title this as. We refer to this type of phenomena as 'volatility' but this apparently has a specific context with regards to statistical phenomena so any ...
- 151
0
votes
0
answers
75
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Minimum of n independent, but not identically distributed inverse Gaussians
I would like to find the probability distribution of the minimum of of n independent, but not identically distributed, i.e. differently parametrized inverse Gaussians. I would prefer an analytical ...
- 13
3
votes
1
answer
1k
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Mean and variance of maximum of normal random variables
I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$.
Note that the $X_i$ are independent, but not identically distributed. That is, ...
- 301
2
votes
2
answers
92
views
Measure that takes samples that is minimized in expectation for a uniformly-distributed random variable?
I am having trouble thinking of a function that operates on a set of samples, that is, single-valued random variables between zero and one, $x_i \in (0,1), i\in\{1,2,...I\}$, and provides a measure of ...
- 187
2
votes
1
answer
6k
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Computing the Hessian of maximum log likelihood function
I am trying to find the Hessian matrix for the maximum log likelihood function given training data ${(xi, yi)}$ for $i=1:N$ with $yi ∈ \left\{+1, −1\right\}$ for each $i = 1,\dots, N$ for the function:...
- 23
2
votes
1
answer
7k
views
Name for (maximum+minimum)/2 and relationship to average?
Is there a common name for $c := \frac{max(X)+min(X)}{2}$? What is the relationship between $\tilde{x} := Avg(X)$ and $c$? What metrics or information can I derive from $\tilde{x}$ and $c$?
If I ...
- 481
7
votes
1
answer
1k
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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)...
- 309
5
votes
2
answers
259
views
Is it possible to obtain more accurate annual extremes predictions from sub-annual data?
I'm looking at various extreme climate variables, such as 50-year or 500-year maximum daily precipitation, using a generalized extreme value (GEV) distribution. The problem with this is that there are ...
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