# Tagged Questions

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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### winsorization - reducing the effect of outliers

I have measured the response times each participant took to respond to 24 items, however, only times of the correct responses of each participant were considered, thus leaving me with distinct number ...
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### Finding local maxima for dummies

Let's imagine we have a matrix with zeros and ones with different ingredients and a binary outcome as well which indicates whether a given meal was good or bad. For example, we've already bought ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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In this thread the first two moments of the two-parameter GPD are given, where the distribution might be defined as $G(y)= \begin{cases} 1-\left(1+ \frac{\xi y}{\beta} \right)^{-\frac{1}{\xi}} & ... 1answer 154 views ### Using extreme value theory to estimate bounds Suppose I have I have a random variable$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$, but assume it is not ... 0answers 207 views ### How to find the$(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)$converges to the Gumbel CDF. Not only did I not ... 0answers 322 views ### Extreme Value Theory: Lognormal GEV parameters Lognormal distribution belongs to the Gumbel maximum domain of attraction, where:$F^{logN}(x; \mu,\sigma)=\Phi\left(\frac{\ln x - \mu}{\sigma}\right)$,$F^{Gum}(x;\mu,\beta) = e^{-\exp\left({-\frac{...
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(...