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Questions tagged [extreme-value]

Extreme values are the largest or the smallest observations in a sample; e.g., the sample minimum (the first order statistic) and the sample maximum (the n-th order statistic). Associated with extreme values are asymptotic *extreme value distributions.*

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Extreme Value Simulation from Copulas with Monte Carlo [on hold]

I'm trying to simulate the tail values from a multivariate distribution using copulas. I'm using Vine Copula package of R to derive the suitable copula for my data and I generate random samples out of ...
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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 ...
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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) ...
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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....
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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)}{(\...
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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)}/...
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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 ...
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How to find the probability curve for maximum/lowest values of a random distribution?

I had asked this question on another forum but they recommended me posting here. This is my problem: I have a continuous variable where I can only measure some points of data and I need to assess the ...
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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 ...
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Fitting Dist GEV: What if the shape parameter is positive but the mean excess plot trend is downward?

Please correct me if I am wrong, when the shape parameter of the GEV is positive, the mean excess plot has a downward trend. But, when I tried to do it with my data, the GEV' shape parameter is ...
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Extreme Value Theory - Can I apply the Hill estimator to block maxima?

I want to apply the Hill estimator for the shape parameter: $ \hat{\xi}_{k,m}^{hill}= \frac{1}{k}\sum_{i=1}^{k}\log \frac{x_{(i)}}{x_{(k)}}\quad 2\leq k\leq m$ where $\{x_{(i)},i=1,...,m\}$ are the ...
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Manipulating samples with Extreme Value Theory

First of all, i'm an electrical engineer and i know very little about statistics and probability. In one of my work, i'm trying to somehow combine(this word might be wrong..) Extreme Value Theory and ...
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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 ...
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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 ...
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Confused: The data length of covariations in a nonstationary GEV

I have established a non-stationary GEV model and expressed the location parameter μ(x) as a function of two covariates (x1, x2) to reflect changing conditions, while the scale and shape parameters ...
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Is there a closed-form solution for ratios of order statistics?

Is there a closed-form solution for the expected value and variance of the ratios between specified order statistics drawn from a large sample from a known parametric distributional family? Actually, ...
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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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How to interpret this mean excess plot?

I have a mean excess plot which looks like this: The "kink" in the blue confidence interval is where we cross from negative treshold to positive. As you can see, it still decreases in linear fashion ...
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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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pooled GEV return level estimation

Let’s suppose I have 4 AMAX time series of 40 years each one, they could be eg wind speeds from 4 meteorological stations in a given region. The time series have been normalised. I want to fit a GEV ...
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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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Are there theoretical reasons for choosing between similar distributions?

I'm interested in estimating the distributions of a few skewed datasets, for example extreme heat, and extreme rainfall. There are many distributions that can be fit to these kinds of data, for ...
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How to interpret the probability density function exceeding one over a finite interval? [duplicate]

If one looks up 'Frechet Distribution' on wikipedia, one will find the following figure in the top-right of the page I was under the impression that the integral of the PDF function taken from ...
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How is this way of rewriting extreme-value problems a simplification?

This question is about pp. 370-374 of Harald Cramer's 1946 Mathematical Methods of Statistics. The author considers a more general question, but for simplicity let us focus on the question of: ...
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$\mathbb{E}[\min (X_{1:n}) + \max(X_{1:n})]/2 = \mathbb{E}[\text{median}(X_{1:n})]$?

Say $X$ is continuous random variable, and we have $n$ iid samples, denoted as $X_{1:n}$. Then can we say the following $$\mathbb{E}[\min (X_{1:n}) + \max(X_{1:n})]/2 = \mathbb{E}[\mathrm{median}(X_{...
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What is the most powerful result about the maximum of i.i.d. Gaussians? The most used in practice?

Given $X_1, \ldots, X_n, \ldots \sim \mathscr{N}(0,1)$ i.i.d., consider the random variables $$ Z_n := \max_{1 \le i \le n} X_i\,. $$ Question: What is the most "important" result about these ...
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Distribution of extremes of measurments considering repeated measures

I'm working with measurements for a large sample set and need to quantify the expected extremes. The measurements obviously have some degree of random noise simply in the measurement itself (for the ...
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Statistics of Extremes: Fitting the GEV distribution with MLE vs L-moments

I created a synthetic series that is supposed to simulate a series of peak discharges in blocks of years in arid catchments. The magnitudes were simulated via the Lnorm dist.: ...
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240 views

what is the correct order of parameters of the GEV dist. for the ks.test in R?

I'm trying to evaluate the ks statistic for the gev distribution with ks.test stat function in R. I read the help a few times and remained puzzled as to what the order has to be and can't find any ...
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Deriving parameters of an extreme value distribution from an initial distribution for finite n

Background: For my research I've been trying to model brittle fracture mathematically. The basic concept is that a brittle materials fail because already existing small defects such as small cracks ...
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114 views

Joint CDF of dependent random variables: is knowing covariance sufficient?

Let $X,Y$ be real-valued random variables, which are dependent. Want: Calculate $\mathbb{P}[\,\min\{X,Y\}\leqslant0\,]$ (without Monte Carlo) Know: I can compute (numerically) $F_X$ and $F_Y$ (the ...
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Approximating the mathematical expectation of the argmax of a Gaussian random vector

Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$. $I$ ...
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139 views

return level in non-stationary case using GPD (POT approach)

I'm doing some extreme value analysis, specifically, using a POT-approach and I'm trying to add some covariates to model excesses. Since I'm quite new in extremes, I'd like to ask for some help to ...
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Estimating EVT for non-i.i.d. data

I have a pnl time series (length more than 10 years) of a large diversified financial portfolio. on which i am trying to estimate VaR based on the method described in the paper : "Estimation of Tail-...
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A question about qqplot

This is my qq plot : Its concave-convex curve so it indicates light tails. But my mean excess plot : is increases which means the tail of the distribution of my data is heavy-tailed. I don't ...
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Maximum Likeilhood estimate of shape parameter of GPD is negative, even though exceedances are positively skewed

I am looking at fitting a Generalized Pareto Distribution (GPD) to extreme events which exceed a certain value threshold for Bilbao waves data. Selecting threshold at c=7.5, resulting in 154 ...
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Maximum likelihood and Gumbel distribution. Does the likelihood have a global maximum?

It appears to me that if I move the mode $u$ more to the negative and increase the scale parameter $\alpha$, one can get always a higher likelihood. If this is true, is there a limit of the likelihood?...
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Decision trees, Gradient boosting and normality of predictors

I have a question regarding the normality of predictors. I have 100,000 observations in my data. The problem I am analysing is a classification problem so 5% of the data is assigned to class 1, 95,000 ...
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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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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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How to calculate the VaR of an equally weighted portfolio of 2 assets using the copula approach?

The R-code procedure in the GARCH-EVT-Copula model estimation I have been able to do the following steps in R: Fit GARCH models to each series. Extract standardized returns. Transform standardized ...
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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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Compare maxima of two Bernouilli experiments

I am looking at the following question -- which has already been solved for the case of Gaussian samples Compare maxima of two Gaussian samples but I am unable to find a similar answer for the ...
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92 views

Using quantiles to estimate the parameters of a distribution: adjusting for unobserved extreme values

I with to estimate the parameters of a specified semi-infinite distributional family based on a sample drawn from that distribution. It is plausible that my sample median converges to the population ...
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156 views

What is the expected maximum value of a gamma distribution, as a function of number of samples?

I have the following situation. I have observations that fill out a gamma distribution. (At least they seem to: the distribution of values of several thousand observations looks to the eye like a ...
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Convert 30 year risk data to 100 year risk data [closed]

I have timeseries of synthetic daily rainfall data for which I want to extract the 100 year flood scenarios using various prediction methods, time-periods and data sources to increase the redundancy ...
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242 views

Distribution of extreme values, case of uniform

Question: For $U_1 , \dots, U_n$ i.i.d. $U \sim \mathrm{unif}[0,1]$, we want to find the asymptotic distribution of $Z_n = n(1-U_{(n)})$ where $U_{(n)} = \max(U_1 , ... , U_n)$ I found this: ...
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Approximation/bound to a_n and b_n in normal maxima to Gumbel

I was reading this which tries to find $a_n$ and $b_n$ such that $$F\left(a_n x+b_n\right)^n\rightarrow^{n\rightarrow\infty} G(x) = e^{-\exp(-x)},$$ where $F$ is the cdf of a standard normal. The ...
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Behavior of mean excess plot

Disclaimer I know that diagnostics with mean residual life plots are extremely liable to subjective interpretation. I was just curious about a consistent behavior I saw running several POT analyses ...
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Generalized Pareto distribution (GPD)

I would like to understand the functional form of the Generalized Pareto distribution (GPD) presented in Wikipedia. My questions are: what is the rationale for replacing $z$ with $\frac{x-\mu}{\sigma}...