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 theory: GPD larger expected value than average

We're using extreme value theory to model tail risks on our portfolio. After we choose the threshold, we fit generalized Pareto distribution to our data over the threshold. The expected value of GPD ...
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Extremal Index of the Dependent Sequence

I am currently studying for my final exam, and am looking at previous years exam and came across this question, can someone please tell me how to do this. The limiting distribution G(x) of the ...
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Theoretical/intuitive question about time-varying Generalized Pareto Distribution

I fitted the GPD to the right tail of nine log return series (I multiplied log returns by -1, so modeling the right tail equals modeling the losses) with a threshold equal to the 95% quantile. Some of ...
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Is it possible to derive a relation between parameters in Poisson process representation of extremes and parameters in GPD model?

I want to derive the theoretical relation between the parameters in a point process model for extremes and the parameters in the GPD model for extremes. I'm following Coles - An introduction to ...
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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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sum of quantile for GEV distribution

Let $q_x$ and $q_y$ be $p$-quantiles for $\text{GEV}_X$ and $\text{GEV}_Y$, where $\text{GEV}_V$ stands for the Generalized Extreme Value distribution associated with sequence of r.v. $(V)_1^n$. That ...
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33 views

How is finding outliers using extreme value theory different from setting threshold on a pdf of normal events?

Suppose we assume that the data follows a Gaussian distribution. We can set a threshold on its pdf to find outliers. How would it be different from setting a threshold on a pdf, fit the exceedances ...
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23 views

Return Level for GEV Distribution

The generalised extreme value (GEV) distribution is a family of distributions defined by 3 parameters; location (µ), scale (σ > 0) and shape (ξ). A GEV(µ, σ, ξ) has cdf: G(x) = exp [−1 + ξ((x − µ)/σ)...
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Predictive modeling of an complex panel of heavy-tailed data

I am struggling to develop a sensible strategy or protocol for the predictive modeling of a complex set of data. Apologies in advance for the indeterminate nature of some of this description but it’s ...
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43 views

Interpreting ACF Plots

I need help interpreting this ACF plot. It was produced on R, where the function acf(ammns) was applied. "ammns" is the annual maxima of rainfall extracted from a dataset of monthly total rainfalls, ...
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274 views

Is the logarithmic transformation sufficient to tame every distribution?

Today I realized a quite known fact. The log transformation of a random variable, drawn from a fat tail distribution, maps into an exponential tail distribution. ...
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277 views

Extreme values in the data

I have a very general statistical question. If a variable has some extreme values, then for the purpose of statistical inferences for example OLS regression, is it better to detect these extreme ...
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89 views

Fit generalised extreme event distribution in MATLAB/R

I have a location-scale t-student distribution $X$ $$ f(x,\mu,\sigma,\nu) = \frac{1}{\sigma} \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi\nu} \Gamma (\frac{\nu}{2})} \left( 1 + \frac{(x - \mu)^2}{\nu\...
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16 views

Test extreme value distribution in Logit Model

Suppose I have a Logit regression with error term. How to test the error term follows an extreme value distribution? I check Logistic Regression in Wikipedia it says The choice of the type-1 ...
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1answer
22 views

Likelihood for dependent data above a threshold

Let $(Y_t)$ a real-valued stationary Markov chain and $u$ some positive threshold. We assume that for $y>u$, $$Y_{t+1}|\{Y_t=y\}\sim\mathcal{N}(\alpha y+\mu y^\beta,\sigma^2 y^{2\beta})$$ I want ...
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51 views

Dealing with Extreme Response Style

My survey data includes a battery of 36 Likert-scale questions, which are the IVs of interest in my analysis. Respondents rank how "alike" they are to an individual being described in each item. ...
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1answer
28 views

Extreme Value Distribution of dependent RV

Let $X_1, \dots, X_n$ be random variables with known identical distribution and covariance matrix $C \in \mathbb{R}^{n\times n}$. How can I model / calculate the extreme value distribution: $$ f(w) = ...
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25 views

Extreme value theory?

I am studying probability and finding hard to understand the following equation. $Pr[P_i \leq \min\{P_s;s \neq i\}]=\int_0^\infty \underset{s\neq i}{\prod}[1-G_s(p)] \, dG_i(p)$ where $P_i$ are ...
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183 views

Have MLE estimators for Generalized Pareto Distribution. Given a known value of $c$, how do I calculate $a$ and $b$ using the provided estimators?

I am doing research into the three parameter Generalized Pareto Distribution $$ f(x|a,b,c) = \frac 1 b\left(1+a\left(\frac{x-c}{b}\right)\right)^{\big(-1-\frac 1 a\big)} $$ for finding VaR and CVaR. ...
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1answer
43 views

Extreme Value Theory - Normalizing constants for Generalized Extreme Value distribution

I'm working on Extreme Values Theory, and I found the following sufficient condition to find the domain of attraction of a distribution and the corresponding normalizing constants: For sufficiently ...
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23 views

Expectation of the minimum of the inverses of a sequence of iid RV

Definitions: Let $\{X_i\}_{i = 1,2,...,n}$ denote a sequence of $n$ i.i.d. random variables (RV), $\mathbb{E}$ the expectation value, $|\cdot|$ the absolute value, and $\min(\cdot)$ the minimum of a ...
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427 views

What is the distribution for the maximum (minimum) of two independent normal random variables?

Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed). Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or ...
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relationship between rapidly varying tails and relatively stable distributions

Suppose a random variable X has cdf $F$ has rapidly varying tail $\overline{F} =1-F$, such that: $$ \lim_{x \to \infty} \frac{\overline{F}(x\lambda)}{\overline{F}(x)}= 0 $$ if $\lambda >1$, and $\...
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80 views

Best hypothesis test for generalized extreme value distributions?

I am trying to compare 3 non-normal distributions with some sort of hypothesis test. Data sets 2 and 3 are subsets of dataset 1 (all data). According to BIC and chi squared fitting in MATLAB, the ...
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51 views

Extreme Value Theory Data Scaling

I have a data set available of almost thirty years of data, with for each month the number of occurrences of a certain event and the total number in the set available. What I would like to compute is ...
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1answer
116 views

joint probability distribution of $k \le n$ order statistics

For $X_i \sim$ iid random variables: For $1\le r_1 < ..<r_k \le n$ integers, I am trying to find the joint pdf of: $$ (X_{(r_1)},...,X_{(r_n)}) $$ where $X_{(r_1)}$ is the $r_1$th largest ...
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126 views

Autocorrelated Inter-arrival Times of Extreme Events

I'm using a bunch of techniques and methods from Extreme Value Theory to analyze my data. I have a time series representing the number of events happening in a given day. The time series is unequally ...
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896 views

Why use extreme value theory?

I'm coming from Civil Engineering, in which we use Extreme Value Theory, like GEV distribution to predict the value of certain events, like The biggest wind speed, i.e the value that 98.5% of the wind ...
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73 views

How to determine how many simulations to run, in order to illustrate “extreme-valued statistics”?

As an engineer trying to learn statistics, I wonder if someone could please recommend references / a statistical method that may assist with determining the number of simulations that need to be ...
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Distribution of the maximum of two correlated normal variables

Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly normal with correlation coefficient $r$. What is the distribution function of $\max(X_1, X_2)$?
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27 views

What statistic to use to measure effectiveness of treatment on fluctuating process

I have a process $R$ that normally does something like a random walk between 0 and 1. I have a set of treatments. I believe that some of the treatments will bias the process $R$ in such a way that, ...
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385 views

Asymptotic distribution of the max (min) of IID binomial variables

I would like to know the limiting distribution when $k \uparrow \infty$ and $k/n \rightarrow \lambda$ of $$ \max(X_1, \ldots, X_k), \text{ where $X_i$ are IID $B(n,p)$}.$$ This is most likely a ...
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166 views

Male and Female Chess Players - Expected Discrepancies at Tails of Distributions

I'm interested in the findings of this paper from 2009: Why are (the best) women so good at chess? Participation rates and gender differences in intellectual domains This paper attempts to explain ...
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644 views

Different quantiles of a fitted GPD in different R packages?

I am performing an extreme value analysis for meteorological data, to be precise for precipitation data available in mm/d. I am using a threshold excess approach for estimating the parameters of a ...
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49 views

Exclude Some samples for calculating CDF

I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and $p&...
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282 views

Fitting a GEV distribution - non-negative only

I am fitting a GEV distribution to some rainfall data, but the software I am using (Matlab and Easyfit) are giving a distribution which includes negative numbers (i.e. negative rainfall). Is there a ...
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93 views

Properties of the minimum of several random variables

I've come across an interesting problem in my research that I don't quite know the answer to. Suppose I have a bunch of random variables: $$ X_1, X_2, X_3, ... X_N $$ They are not identical but they ...
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597 views

Using bootstrap to obtain sampling distribution of 1st-percentile

I have a sample (of size 250) from a population. I do not know the distribution of the population. The main question: I want a point estimate of the 1st-percentile of the population, and then I want ...
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56 views

Is margin of error truly valid at extreme proportions? Such as 1% agreement, mere traces of data

Survey margin of error contracts as the proportions become more extreme. Its validity and applicability in such cases has always concerned me, but I suppose much depends on the context. Where we have ...
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123 views

Missing data on Extreme Value Analysis

I am analyzing (extreme value analysis) the dataset which contain daily rainfall over 100 years of a single location. However there are around 500 missing values on the whole dataset. In this case the ...
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2answers
158 views

Extreme Value Theory and heavy (long) tailed distributions

I'm analyzing data about which I have a strong suspicion that it is self-similar (Hurst parameter ranging from 0.60 to 0.78 depending on estimation method and sample sequence). I also observe high ...
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1answer
33 views

Estimating costs with extreme values

I am trying to estimate health care costs and I was wondering what the standard practice is for extreme values? By extreme values I mean I have a large portion of my costs being zero and a small ...
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120 views

Extremal serial dependence

As part of my analysis of heavy-tailed time series of company returns, I would like to check whether extreme returns exhibit serial dependence, i.e. if extreme events are followed by extreme events. ...
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663 views

Moments of the two-parameter generalized Pareto distribution (GPD) needed

In this thread the first two moments of the two-parameter GPD are given, where the distribution might be defined as $G(y)= \begin{cases} 1-\left(1+ \frac{\xi y}{\beta} \right)^{-\frac{1}{\xi}} & ...
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Using extreme value theory to estimate bounds

Suppose I have I have a random variable $X$ that I know is doubly bounded on support $[0,\theta]$ but I dont know $\theta$ (we don't know anything on the distribution of $X$, but assume it is not ...
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How to find the $(a_n,b_n)$ for extreme value theory

In the solution to this question (Extreme Value Theory - Show: Normal to Gumbel), the OP asked for the sequence $(a_n, b_n)$ such that $\Phi(a_nx+b_n)$ converges to the Gumbel CDF. Not only did I not ...
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Extreme Value Theory: Lognormal GEV parameters

Lognormal distribution belongs to the Gumbel maximum domain of attraction, where: $F^{logN}(x; \mu,\sigma)=\Phi\left(\frac{\ln x - \mu}{\sigma}\right)$, $F^{Gum}(x;\mu,\beta) = e^{-\exp\left({-\frac{...
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Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = P(...
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1answer
358 views

Required: Method of moments fitting routine for the two-parameter generalized Pareto

I am currently using the evd package which fits a two-parameter GPD by maximum likelihood. Since in small samples the MOM is superior to the ML estimation I'd like ...
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GEV of Normal Distribution and relationship of the parameters

My question goes on Extreme Value Theory for the Normal distribution (www.math.ethz.ch/~embrecht/RM/chap7.pdf): Which type of GEV (Generalized Extreme Value) distribution does the Normal ...