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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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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445 views

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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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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172 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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123 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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123 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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43 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 ...
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47 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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40 views

How to estimate Extreme value distribution parameter

Assume that I have non-negative Gamma random variables $\{X_i,i=1\dots n\}$ and I want to find $M_n=\max\{ X_i\}$. I want to apply generalized extreme value distribution (GEV), but how to find $\mu$, ...
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Find Points that Break a mutli-valued Relationship

I am looking for a suggestion on how to answer a question, or what to read more about. I am sorry if I do not sound professional, I am not. First a contrived example. I have a table of 50 states in ...
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164 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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36 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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44 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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1answer
68 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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69 views

How to find yearly return level form hourly data?

I have hourly values from which I extract every data over a threshold (Peak-over-threshold method). I then filter those extreme values so that i only get one value in a given 48 hours period. This ...
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1answer
28 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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60 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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177 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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104 views

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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63 views

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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126 views

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) = ...
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437 views

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) = ...
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172 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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51 views

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 ...
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68 views

expected shortfall and value-at-risk

I once read a R example of computing Value-at-Risk and expected shortfall as follows ...
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47 views

Confidence interval in EVT

I'm working through a Power-Point presentation about extreme value theory with application to finance. My question is about a technique to calculate the confidence interval of a $k$ $n$-block return ...
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Correct scaling factor going from monthly to yearly data

I'm trying to compute the probability of an extreme event, using extreme value theory on my data. One of the typical problems with this kind of computations is the lack of sufficient data, which also ...
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179 views

Given the location and scale parameters of a Gumbel distribution for variable X, how does one calculate the mean and variance of X^2?

I am working with predictive models for wind speeds, which have been given as Gumbel distributions. I need to convert the wind speeds to wind pressures using the formula: $Pressure = Density * ...
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42 views

Modelling the tail only

I'm trying to model a real-world random variable that behaves approximately as a Gaussian, so a Normal distribution fit is reasonable but far from perfect. However, I only care about its tail, that ...
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158 views

Extreme value simulation with Monte Carlo

I would like to seek your help with some questions to simulating extreme values. For example, I have written the following R code to generate QQplots for a normally distributed data, varying the size ...
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90 views

Best method to fit a GEV distribution with generalised linear modelling of parameters?

I need to fit a generalised extreme value distribution to my data but I want the ability to perform generalised linear modelling of the parameters, particularly the location. Can anyone recommend the ...
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51 views

Extreme value distribution for multivariate normal

I have a series of data sets. Each data set represents a measurement in 3D space relative to a global origin. I want to model the extreme values of my data. If I were to calculate the extreme radius ...
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57 views

Lévy stable vs. extreme value distributions

I'm trying to understand the advantages (if any) of employing the Generalized Extreme Value distribution (GEV) vs. a stable distribution in the context of understanding the probability of crossing a ...
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Repairable system and the sum of GEV random variables

We know that $X\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$ and $Y\sim {\mathrm {GEV}}(\alpha ,\beta ,0)$ then $X+Y\sim {\mathrm {Logistic}}(2\alpha ,\beta )$. I am wondering, what will be $X+Y+Z$ ...
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Conceptual or mathematical motivation for the three extreme value distribution types?

What motivates, justifies, gives rise to the differences between the Gumbel, Fréchet, and Weibull distributions? Glen_b's comment indicates that they are distributions for extreme values generated by ...
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228 views

Fitting a linear model with non gaussian noise

I am trying to evaluate the elasticity of prices of some goods. I am concerned about the gaussianity of the noise in the prices. With non gaussianity I am referring to the non existence of the ...
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122 views

expected lowest value of 10 normally distributed values

Consider 10 values that follow a standard normal distribution. What would you expect to be the lowest value? I tried to simulate this problem in R. I basically just simulated 100000 standard normal ...
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1answer
202 views

Fitting a probability distribution to non i.i.d. data? [closed]

I have temperature time series data that I have determined is not independently and identically distributed (from looking at the autocorrelation plots and Ljung-Box tests). However, I am still able ...
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91 views

Compare maxima of two Gaussian samples

Suppose $X$ and $Y$ are both normally distributed, with $X \sim \mathcal{N}(0,1)$ and $Y \sim \mathcal{N}(c,1),$ where $c > 0$. Consider $n$ independent draws of both $X$ and $Y$. As $n \rightarrow ...
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97 views

Distribution of Extreme Spread for n, sigma

Simple form provided by WHuber: What is the distribution of the diameter of n points in the plane drawn iid from a bivariate Normal distribution? (Diameter is the greatest distance among any pair of ...
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95 views

Joint cdf of extreme values

A die is rolled twice, $X_1$ : the minimum value to appear in the two rolls $X_2$ : the maximum I would like to derive $\ F_{X_1,X_2}(x_1,x_2)$. I know that that the CDF of $\ X_1 $ = $\ 1- ...
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204 views

Extreme value theory for count data

I am aware of extreme value theory for continuous distributions. I need to fit an extreme value distribution to the maximum observation of number of events on a day, per month. This seems to be the ...
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OLS robust to outliers

I am facing the following problem: I have a training sample and estimate a model on that training sample. My model is simply OLS: $y_t = a + \beta x_t + \varepsilon_t$. The model is estimated on ...
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175 views

What is loc parameter in GPD distribution in POT package for R?

I fitted the Generalized Pareto distribution (GPD) using the POT package in R. The fitted object provides shape and scale ...
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302 views

Two player dice game probability

$A$ and $B$ play a dice game where a player wins if their score is higher. $B$ wins if their score is equal. What is the probability of $A$ winning if both the players roll their dice $n$ times and ...
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246 views

Convergence in distribution of the maximum of a sequence of random variables

How to find the following: Let $X_1$, $X_2$, $X_3$,..., $X_n$, be i.i.d with chi-square distribution with one-degree of freedom. Find $a_n$ and $b_n$ such that $ a_n(\max_i X_i - b_n)$ converges in ...
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Tail bounds on Euclidean norm for uniform distribution on $\{-n,-(n-1),…,n-1,n\}^d$

What are known upper bounds on how often the Euclidean norm of a uniformly chosen element of $\:\{-n,~-(n-1),~...,~n-1,~n\}^d\:$ will be larger than a given threshold? I'm mainly interested in bounds ...
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Forming an unbiased estimator of the maximum of several parameters, given independent estimators of each parameter?

Say I have $K$ independent normals, $X_i \sim \mathcal{N}(\mu_i, \sigma_i)$ for $i = 1,...,K$. How can I form an unbiased estimator of $\max_i \mu_i$ using $X_i$'s?
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89 views

What is the meaning of McFaddens Axiom: Irrelevance of Alternative Set Effect?

On page 110 of McFadden,1973 - Conditional logit analysis of Qualitative Choice Behavior, Frontiers in Economics, ed Zarembka, New York: Academic Press, pp. 105-142 the following three Axioms are ...
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401 views

Asymptotic probability concerning the largest absolute value in an iid Gaussian sample

Let $v[n]$ be a vector $N$ $iid$ gaussian samples, of ~$N(0,\sigma^2)$ Also, let $v_{max}$ denote the maximum absolute value of all the samples given. (That is, if I took the absolute value of all the ...