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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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How to deal with outliers in panel data? [closed]

When we have cross-sectional data, we can easily detect and remove outliers. But how should one approach outliers when we are dealing with panel data? Since we have $i$ entities and $t$ times periods, ...
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How to understand intuitively the CDF formula for the maximum statistic of three iid rv’s? [duplicate]

Given that all three iid rv’s ($X_1, X_2, X_3$) have CDF $F(x)$, the formula for the CDF $G(y)$ of the largest rv ($Y=X_i$) among the three is: $G(y)=P(X_1 \leq y) \cdot P(X_2 \leq y) \cdot P(X_3 \leq ...
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Derivation of a dynamical Generalized Pareto distribution

I'm currently reading a paper for my master thesis on the tail index estimation for asset returns using the peak over threshold method. In this paper the authors introduce the cumulative distribution ...
data_science_101's user avatar
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Distribution of a random variable conditional on its being a maximum or not

Consider the random variables $\epsilon_1,\dots, \epsilon_D$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Assume they are continuous. Let $$ Y=\sum_{d=1}^D d\times \mathbb{1}\{\...
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Multinomial Logit Extension

The derivation of the multinomial logit probabilities depends on the difference of two Type 1 extreme value (Gumbel) random variables following a logistic distribution. We say the unobserved utility ...
Adarsh Nayak's user avatar
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How do you determine an appropriate block length for calculating "block maxima" for GEV distribution?

I have some time series data spanning 30+ years and I am trying to do some extreme value analysis. Major disclaimer: I am not a statistician so I feel that I am wading into waters beyond my area of ...
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Validity of bootstrapping for estimation of annual maxima distribution

I am working with a large timeseries (millions data points) spread across 5 years from which I would like to estimate the annual maxima distribution and subsequently a quantile of this distribution. ...
Matthias's user avatar
7 votes
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Estimation of a uniform distribution corrupted by Gaussian noise

Problem definition I have a dataset composed by $m$ observations $y^{(1)},\dots,y^{(m)} \in \mathbb{R}^2$ generated as follow \begin{equation*}\begin{aligned} y &= z + v \newline z & \sim\...
matteogost's user avatar
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How should I best to use reported stats on the Tippy-top?

Suppose I have a large population, in the millions, drawn from some underlying distribution, which we will take as a member of a known distributional family with unknown parameters. Assume the ...
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Bootstrapping moderately extreme quantile regression

Let $(Y_1, X_1), \dots, (Y_n, X_n)$ be iid sequence drawn from $F$. For a fixed $q\in (0,1)$, consider the linear q-quantile regression $Q_Y(q|x) = \beta_qx$, where $Q_Y(\cdot\mid x)$ is the ...
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Two-sample test of difference in probability mass at the extremes of the empirical distributions

I am running an experiment that will generate a dependent variable (DV) in two treatments, T1 and T2. One of the hypotheses I want to test is whether the distribution of the DV in T1 has more mass at ...
hangingprawns's user avatar
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How to do hypothesis testing for Minimum value?

I have a sample with a size of n=100, and I want to show that the minimum value of the underlying distribution is not less than a certain threshold, with a confidence level of 95%. The distribution of ...
Joe the Second's user avatar
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Finding the temperature value that gives optimal value

I'm trying to analyze some sleep data from kaggle (this example data does not have correct temperature data but the actual data I will use in the future will have precise temperature) to try to find ...
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Threshold choice for Peaks-Over-Threshold

I'm trying to estimate equivalent performances at different events, using Peaks-Over-Threshold from Extreme Value Theory. The challenge is to find the threshold and preferably with same number of ...
Daniel Westergren's user avatar
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Extreme Value Analysis - Nonrandom/Preferential Sampling

I am doing an extreme value analysis (EVA) but there is a nuance in my problem that I believe is not addressed in extreme value theory. I have not been able to find information about this in textbooks ...
In the Limit's user avatar
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Separating components of a likelihood maximization

Apologies for the naive question, but I have a problem I would like to solve. Suppose I have a two dimensional likelihood of the form $L \propto \exp\{-\frac{1}{2}\} \begin{bmatrix}x & y\end{...
Fellow99's user avatar
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How to find the MGF of the max of a set of i.i.d. exponential random variables

As the title suggests, I would like to find the MGF of the max of iid exponential random variables. Assume $Z=\max(x_{1},...,x_{n})$, where $x_{i}$ is distributed as exponential($\beta$) and has pdf $\...
stats6895997's user avatar
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Max of the running average of the kth through nth elements for a given probability distribution

This question is based slightly on https://www.reddit.com/r/AskStatistics/comments/16bqit0/calculating_probability_when_phacking_is_allowed/ Given a variable $X$, let $A_j$ be the average of $X_1$ ...
Barry Carter's user avatar
11 votes
1 answer
221 views

Distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$ when $X_i$'s are i.i.d $\text{Exp}(1)$

Suppose $(X_n)_{n\ge 1}$ is a sequence of independent Exponential random variables with mean $1$. I am trying to find the distribution of $\min_{j\ge 1}(X_1+X_2+\cdots+X_j)/j$. Simulation suggests the ...
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How to find an "upper margin" for data on visits

For a handful of store locations, I have data on each entrance and exit time. I have counted the total num of people at a store at any given minute. I am trying to find out the values for which the ...
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Extremal function as defined in the paper Graphical Models for Extremes

The following is a question based on the paper Graphical Models for Extremes by Engelke and Hitz. For our purposes, exponent measure is a measure defined on $[0,\infty)^d - \{0\}$ with a density $\...
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Definition of exponent measure (extreme value theory)

Let $F$ be a distribution function on $\mathbb{R}^2$, and let $U_i$ be the left continuous inverse of $\frac{1}{1-F_i}$, where $F_i$ is the marginal distribution of $F$. In my textbook, there is the ...
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Proof of convergence of distribution of order statistics

Let $X_i$ be i.i.d with distribution $F$ such that for some $a_n>0$, $b_n \in \mathbb{R}$. $F^n(a_n x +b_n) \to \exp(-(1+\gamma x)^{\frac{-1}{\gamma}}$ Define $X_{1,n} \leq \ldots \leq X_{n,n}$ to ...
Phil's user avatar
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6 votes
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Running maximum of $\sum_{1\leq k\leq n} X_i$ for Cauchy random variables $X_i$

Suppose $X_i$ are $\mathrm{Cauchy}(0,~\gamma)$ IID RV's and let $S_n=X_1+\cdots+X_n$ be their sum. Does an expression exist for the CDF of the running maximum up to an index $1 \leq k \leq n$? Edit: ...
user169291's user avatar
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Peak-Over-Threshold for multiple weather stations in the US

I'm trying to make an Extreme Value Analysis (EVA) of maximum and minimum temperature data from all over the US and try to fit a Generalised Pareto Distribution (GPD) according to theory after ...
Nemtexer's user avatar
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Clarification on definition: $\Gamma$-varying function (Extreme value theory)

In Extreme Values, Regular Variation and Point Processes, there is the following definition: A non-decreasing $U$ is $\Gamma$-varying if $U$ is defined on an interval $(x_l, x_0)$, $\lim_{x \to x_0^-} ...
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Tail of the maximum of a time-varying Poisson-GP marked process

Consider a time-varying version of the Poisson-GP marked process on the real line as commonly used in Peak Over Threshold (POT) modelling of a variable $Y$. More precisely we have a given time-varying ...
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$1-F$ is rapidly varying if and only if there exists $b_n$ such that $\frac{\max X_i}{b_n} \to 1$ in probability

The following is a problem from Extreme Values, Regular Variation and Point Processes by Resnick. We will say $1-F$ is rapidly varying as $x \to \infty$ if $\lim_{t \to \infty} \frac{1-F(tx)}{1-F(t)} =...
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Empirically estimating extremal coefficient using minima of Fréchet margins

I recently came across a paper which uses the following formula to empirically estimate the extremal correlation coefficient $\chi_{ij}$ between two variables $x$ and $y$ as follows: $$ \chi_{xy} = \...
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Recreating Heffernan-Tawn Fig. 6 in R: samples from multivariate extreme value distribution

I am trying to recreate Figure 6 from Heffernan-Tawn (2004) A conditional approach for multivariate extreme values using the same datasets as used in the paper. The original time series data of air ...
ThreeOrangeOneRed's user avatar
3 votes
1 answer
62 views

Does this approach to simulation for survival analysis, of breaking the analysis into deaths versus survivors, appear reasonable?

I've spent last several weeks learning about survival analysis, see one of the last posts at How to simulate variability (errors) in fitting a gamma model to survival data by using a generalized ...
Village.Idyot's user avatar
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Statistical assessment of block size for bootstrapped distribution fitting

I have a set of intensities from unordered independent events (with no date or timestamps), many of which constitute extremes, and I want to generate an extreme value distribution. The only ...
jeremy's user avatar
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Gumbel distribution conditional on exceeding a threshold

In Heffernan and Tawn's 2004 paper, they describe a procedure to sample multivariate data, conditional on one variable ($Y_i$) being extreme. The idea is that $Y_i$ is extreme if it exceeds some ...
ThreeOrangeOneRed's user avatar
1 vote
1 answer
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If $F^n(b_n x) \to e^{-x^{-\alpha}}$, $b_n x \to x_0$ where $x_0 = \sup \{x \colon F(x) < 1 \}$

Let $X_n$ be i.i.d with common df $F$. Let $M_n = \max (X_1, \ldots, X_n)$. Suppose $P(b_n^{-1} M_n \leq x) = F^n(b_n x) \to e^{-x^{-\alpha}}$ weakly, where $x > 0$ and $\alpha > 0$. Let $x_0 = \...
Phil's user avatar
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How to simulate variability (errors) in fitting a gamma model to survival data by using a generalized minimum extreme value distribution in R?

As shown below and per the R code at the bottom, I plot a base survival curve for the lung dataset from the survival package ...
Village.Idyot's user avatar
4 votes
2 answers
277 views

Does the following distribution converge to anything?

Consider the following process for generating a random sample: Sample $X_1, X_2, \dots, X_n \sim \mathcal{N}(0,1)$ Compute $M = \max\limits_i |X_i|$ Scale the values to get $Z_i = X_i / M$ Can we ...
Davis Yoshida's user avatar
2 votes
1 answer
48 views

How to assign reasonable scale parameters to randomly generated intercepts for the Weibull distribution?

This is a follow-on to post Correctly simulating an extreme value distribution for survival analysis?, as I work towards adaptation of that code to the Weibull distribution. In the below code I ...
Village.Idyot's user avatar
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Resource recommendation for extreme value theory

I'm look to learn about extreme value theory, starting from univariate case and then moving onto the multivariate case. I have tried the textbook by de Haan, but I'm constantly lost trying to read the ...
1 vote
1 answer
233 views

Correctly simulating an extreme value distribution for survival analysis?

In the image and per the code at the bottom of this post, I plot survival curves for the lung dataset from the survival package using a fitted exponential ...
Village.Idyot's user avatar
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0 answers
33 views

Identifiability of a bivariate normal distribution with identified minimum

I am suffering from to understand a proof of a paper. (Nádas, Arthur. "The distribution of the identified minimum of a normal pair determines' the distribution of the pair." Technometrics 13....
MinChul Park's user avatar
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Is modeling the extreme value of a distribution a basic probability result?

I was reading briefly about the field of EVT - extreme value theory, and the associated distributions that arise from modeling the maximum of a finite sample. It's not quite clear to me the nature of ...
AdamO's user avatar
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5 votes
1 answer
279 views

Is every probability distribution also the distribution of the maximum of several i.i.d. random variables?

I found the following result used in this paper, but it was just claimed without proof and it seems extremely strong to me, so I would like a proof, or at least a reference, of a proof. Let $D$ be ...
AspiringMat's user avatar
1 vote
0 answers
35 views

How to calculate Gumbel with LMoments and GEV with method of moments

I need to calculate the values for certain return periods of a flood event (up to 5000). It has to be GEV with method of moments and Gumbel with L-Moments. But I am not sure about how to calculate ...
Ben_1801's user avatar
1 vote
0 answers
56 views

Definition p-value and find p-value in practice

I have a problem that I can't solution. Let $\mathbf{X}=\{X_1,X_2,\ldots,X_n\}\sim\mathrm{Uniform}(0,\theta)$ and we have $H_0:\theta=\theta_0$ and $H_1:\theta>\theta_0$. We reject the $H_0$ when $...
Samvel Safaryan's user avatar
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Find maximum of bimodal posterior pdf

can you help find the maximum (analytically) of the following posterior pdf? $p(\theta|x) = \frac{\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta-x)^2} + \frac{1-\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta+...
st7488's user avatar
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2 votes
2 answers
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Calculating probability related to maximum of random variables

Let $X_1, X_2, \cdots, X_n$ be non-negative continuous iid random variables. The goal is to find the probability: \begin{align*} \Pr(\max_{k+1 \leq i \leq j } X_i < \max_{1 \leq i \leq k }X_i) \end{...
Math Universe's user avatar
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What can be concluded when standard deviation plus mean exceeds largest value?

The sum of the mean and standard deviation of a non-normal distribution can exceed the value of the largest sample. For a good explanation of why, see Can mean plus one standard deviation exceed ...
jsbox's user avatar
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1 answer
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How to choose the wanted root of the maximum likelihood function when there are multiple roots?

I need to estimate a parameter of a distribution but I don't have an explicit estimator. I decided to do a partition of the interval range for the parameter and use the newton-raphson method to find ...
Rui Gonçalves's user avatar
2 votes
1 answer
139 views

CDF of max of $n$ cauchy variates

Suppose $X_1,X_2,\cdots,X_n$ are iid copies of a standard cauchy variate with pdf $$ f(x)=\frac{1}{\pi(1+x^2)},0<x< \infty. $$ Define: $$ Y=1+ \underset{1 \leq i \leq n}\max X_i.$$ I want to ...
AgnostMystic's user avatar
2 votes
0 answers
73 views

Calculating confidence Interval for a return time curve, via non-parametric bootstrapping

I have some precipitation data (yearly extremes), which I have fit with a Gumbel distribution (CDF), from which I have calculated a return time distribution. I want to calculate the 95% confidence ...
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