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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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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16 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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24 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
24 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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23 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
22 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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39 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. ...
2
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1answer
44 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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1answer
85 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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48 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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81 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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1answer
208 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) = ...
5
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1answer
95 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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0answers
33 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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39 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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40 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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17 views

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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2answers
68 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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1answer
38 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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28 views

Statistical Blockade - Estimating Rare Event Statitistics

In the world of microelectronic design and simulation where the technology gets smaller and smaller (a few nanometers nowadays), the high density and replication of some of our circuits require us to ...
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1answer
85 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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20 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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19 views

Smallest Spectral Norm / Deviation Inequality

Consider $A_{m \times n}$ be an i.i.d. random matrix with finite first to fourth moments. There is a good number of asymptotic and non-asymptotic results regarding the spectral norm of $A$, $\|A\|_2$, ...
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40 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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1answer
46 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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42 views

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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77 views

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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63 views

How to model zero-inflated continuous response using categorical predictors - preferably resulting in multiplicative parameters

I'm having trouble finding a suitable model for predicting the AVG value (revenue in cents) of a single click on a product on a large e-commerce site. (assuming a click leading directly to a purchase ...
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1answer
144 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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1answer
107 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
143 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 ...
5
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1answer
84 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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1answer
88 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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82 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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1answer
137 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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85 views

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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1answer
124 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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1answer
263 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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1answer
171 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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164 views

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 ...
3
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55 views

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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80 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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1answer
326 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 ...
2
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1answer
357 views

What do I need to consider when using the Hessian to compute S.E.'s?

I use optim() in R to do a lot of MLE. I've used the approach for a lot of problems, but the one I'm working on right now consists of fitting the parameters of the generalized extreme value ...
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1answer
849 views

Fitting GEV to non-stationary time series of extremes (general stationarity question?)

I'm fitting the generalized extreme value distribution (GEV) to a series of annual maxima of variable $X$. $X$ exhibits a linear trend. When I fit the GEV to $X$, I think I have the choice to Use ...
3
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1answer
203 views

Expected value of latent utility in logistic regression

I'm looking for an analytical expression for the expected value of the latent utility in a logistic regression. Setup: There are two choices indexed by $i \in \{0, 1\}$ with associated utilities ...
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1answer
30 views

How to test the significance of increase in sample interval range(s)?

Suppose we have two samples of a variable taken under different conditions: e.g. A1 without medical treatment and A2 after medical treatment. These are not necessarily normally distributed. Suppose A1 ...
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237 views

Confidence intervals for extreme value distributions

I have wind data that i'm using to perform extreme value analysis (calculate return levels). I'm using R with packages 'evd', 'extRemes' and 'ismev'. I'm fitting GEV, Gumbel and Weibull ...
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43 views

Non-Analytic extrapolation

I have some samples of a stable real-world process. Its is polymodal, and does not cleanly fit any of the "textbook" analytic distributions. I need to make very accurate estimates of the maximum ...
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180 views

Probability of exceedance and reliability of a sample range estimation

Consider that $P$ is the water pressure coming out of a valve $A$. Let $P_{dif}$ be the difference between the maximum and the minimum pressure of valve $A$: $$P_{dif}≔P_{max}-P_{min}$$ Now, what I ...