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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15 views

Criteria to handle level shifts in time series of annual daily flows for extreme value analysis

Given a time series of maximum daily river discharges that covers a number of years, suppose that it has one level shift at one point (figure below is only indicative and has four shifts!). ...
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How to estimate the maximum value from a set of data with errors?

Say I have a set of n measurements. The measurement process has a known error. I can't assume that the true values being measured follow a normal distribution. How can I estimate the actual maximum or ...
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(Non-limit) distribution of maxima from different univariate, discrete and stationary time series

Motivation: I'm currently studying the convergence of maxima from simulated time series to max-stable distributions, and in order to do so, I want to better understand the penultimate distribution of ...
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Estimating the blockchain mining time for $N$ nodes

I am trying to simulate a set of times for the below problem. There are N nodes. Each node generates a random number($R$) in the range $[0,K]$ per second. Guess the time it takes by each node to ...
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Suppose $max\{a_i\}_{i=1}^{Rn}\overset{p}{\rightarrow} a_0$, where $a_i$ are i.i.d.r.v.. Are there any results on its rate of convergence?

Suppose $max\{a_i\}_{i=1}^{Rn}\overset{p}{\rightarrow} a_0$, where $a_i$ are i.i.d. random variables, $a_0$ is a constant and $R_n\rightarrow\infty$ as $n\rightarrow\infty$. Are there any results on ...
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R - MLE of modified Champernowne density , using the “nlm” function

I've come across an article (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=704903), in which author wrote about maximum likelihood estimates of parameters in the so called modified Champernowne ...
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38 views

Relationship between the number of moments and Tail of the distribution?

While studying about kurtosis and extreme value theory, I came across the concept of tails of the distribution. So I wanted to ask that why is it such that distribution with higher number of moments ...
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68 views

How to fit distributions to data in R?

I have 6 sets of Volume(v) & Duration(d) data. I have fitted a quite few distributions to the data such as Weibull, Gamma, Log-Normal, Exponential, GEV, Pareto, Log Logistic, Poisson, and GP. This ...
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28 views

Distribution of minimum distance in a iid Gaussian sample

$X_1,...,X_n$ denotes an iid sample with the same Gaussian distribution. I am interested in the distribution of the following quantity. We first pick $i \in [n]$ Then we extract $j^* \in argmin_{j\...
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Is power law distribution for extreme event special like normal distribution?

The power-law distribution is defined as below in Wikipedia article: The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. $$ \mathrm{Pr}[X>x] \sim x^{-...
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If the density functions $f_1, f_2$ each are in domains of attraction $MDA(\xi_1)$ and $MDA(\xi_2)$, what can we say about $0.5f_1+0.5f_2$?

My question is about the maximum domains of attraction $MDA(\xi)$ from extreme value theory. I would like to be able to say statements such as "since $f$ and $g$ both are in $MDA(\xi)$, $f+g$ is also ...
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Distribution of a sequence of maximums generated using i.i.d. Normal variables

I am trying to think about the distribution of a random process. Here's how you would generate the sequence: for each sample of size k (sampled from i.i.d. Normal R.V.s), we find the maximum, and let'...
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What is the distribution of max-min for a Gaussian distribution

For a process N(t), where at any instance of t=T0, the distribution of N(T0) is Gaussain with mu=0: What is the distribution of max(N(t))-min(N(t))? From my simulation, it has some non-zero positive ...
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Distribution of the second minimum of a set of random variables

Given $n$ i.i.d. random variables $X_1,...X_n$, what is the distribution of the second smallest value ? I know from this question that CDF of the minimum value is $1 - (1-F(x))^n$ where $F(x)$ is the ...
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Estimate return levels for non-annual data

Apologies if this has been asked before - I have looked but could not find anything. I intend on estimating return levels for average annual temperature data using a number of different distributions....
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Optimize black box function with one of the output as constraint

I have used deep learning to obtain an objective function (or black box) which I need to optimize to get max output. So the inputs are a1,b1,b2,...,b8 while outputs are x,y and z = x^2/y. I need ...
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Unbiased Estimator of Largest Mean of Two Normal Distributions

Given samples from two normal distributions: $X_i \stackrel{iid}{\sim} \mathcal{N}(\mu_X, \sigma_X)$ for $i = 1,...,n$ $Y_i \stackrel{iid}{\sim} \mathcal{N}(\mu_Y, \sigma_Y)$ for $i = 1,...,n$ How ...
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MLE for the maximum of n values that are observed only with noise

Suppose $x_1, ..., x_n$ is a fixed set of real numbers. Let $\epsilon_1, ..., \epsilon_n \sim N(0, \sigma^2)$ be i.i.d. with known $\sigma^2$, and suppose we get to observe only $z_i = x_i + \...
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35 views

Is there something conceptually wrong with calculating the mean of minimum values?

Assume I have distance values (e.g. how close an animal gets to a city) for each individual in my sample. So, for each individual animal, I will have a minimim distance. Next, say I have 4 categories ...
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KL divergence for Generalized Extreme Value distribution

I have found a derivation for the Kullback–Leibler divergence between 2 Gumbel distributions here: http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf on page 64 That document also has a ...
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Argmax of betas

Let $\theta_i \sim \text{Beta}(\alpha_i, \beta_i)$ for $i \in I$. What is the distribution of $i^\ast = \operatorname*{argmax}_{i \in I} \theta_i$? Let $[X]$ denote the CDF of a random variable $X$. ...
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Maximums of two exponentials

The wikipedia page for the exponential distribution states that for $X_1, X_2, \dots X_n$ independent exponentially distributed with rate parameters $\lambda_1,\lambda_2,\dots,\lambda_n$, the index of ...
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Looking for … the Generalized Fisher-Tippett distribution

so I would like to use a parametric model that smoothly interpolates between the 3-parameter Weibull, the 3-parameter Frechet and the 3-parameter Gumbel. Just like the Fisher-Tippett distribution (a....
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Expectation of the minimum of a continuous random variable $X$ and a discrete random variable $Y$

Let $X\sim Exp(1)$ and independently let $Y$ have the pmf $P(Y=k)= p$, $P(Y = \infty) = 1-p$, where $k < \infty$. I'd like to calculate $\mathbb{E}(Z)$, where $Z = \min(X,Y)$. Usually, we tackle ...
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Accounting for extreme events in machine learning models

I am researching ways to account for extreme or anomalous events in predictive models. For example, if I am predicting revenue or consumer demand, what are some ways to account for events, like ...
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Probability of failure set

I want to prove the following theorem, which in general allows us to compute the probability of a certain set even if it contains no observations (which is all too common in extreme value analysis of ...
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Using Gumbel distribution to approximate distribution of sample maximum — formulae for the parameters?

Suppose you have an observable sample $X_1,...,X_n \sim \text{IID } F_X$ which has a right-tail that decreases sufficiently rapidly to apply the extreme-value theorem (e.g., a normal distribution) to ...
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31 views

Random Sampling from Farlie-Gumbel-Morgenstern bivariate exponential distribution

I would like to obtain an algorithm for generating iid samples from Farlie-Gumbel-Morgenstern bivariate exponential distribution (as described in the book by Johnson and Kotz as Gumbel's Model II ...
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Gaussian closest to a maximum of Gaussians

Let $X_i \sim \mathcal{N}(\mu_i, \sigma_i)$ be independent, normally-distributed random variables. Let $$Y = a + b \max_i X_i$$ where $a \in \mathbb{R}$ and $b \in (0, 1)$. Which Gaussian ...
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Parametric model closed under translation, contraction, and maximum

Is there a nontrivial parametric model that is closed under translation, contraction, and maximum? That is, does there exist a nontrivial parametric model $\mathcal{M}$ such that $$\forall i \in I : ...
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190 views

Minimize asymptotic variance of fintely many estimates

Let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $f:E\to[0,\infty)^3$ be a bounded function with integrable Euclidean norm on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\...
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Average of already aggregated data

I have mean, variance, second moment, percentiles, maximum and minimum values of a variable X for each time interval in which I divided my experiment. Does it make ...
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29 views

How error affects maximum detection?

I have a discrete function $f$ which lies over a certain domain $X$. My goal is to find the value of $X$, $x_{max}$, for which the function is maximum. I have opted for a simply search: using numpy ...
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101 views

Compute which of a finite number of integrals is minimal (not interested in the actual value of the integral)

Let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $f:E\to[0,\infty)^3$ be a bounded Bochner integrable function on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ ...
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Variance of $k$th order statistic of normal vector [duplicate]

Let $Z \sim \mathcal{N}(0, I)$. Let $Z_{(k)}$ be the $k$th order statistic of $Z$. Is it true that $\text{Var}(Z_{(k)}) \to 0$ as $n\to \infty$ for $1 \leq k \leq n$? Any estimate on the rate? What ...
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Question about GEV

I'm doing some analysis involving rectangular pulse processes. Suppose for each process {Xi} that X changes after equal so-...
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Choosing parameters for Clifton Extreme-Value-Theory(EVT) novelty detection technique

I was reading your Clifton et all artibles about applying Extreme-Value-Theory(EVT) for novelty detection, eg: http://www.robots.ox.ac.uk/~davidc/pubs/mevt2010.pdf and it looks like one needs to ...
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70 views

Limiting distribution of maximum of i.i.d. Gaussians with decreasing variance

Consider a random vector $X^{(m)} = (X^{(m)}_1,\dots,X^{(m)}_m)$ where, for fixed $m$, the elements of $X^{(m)}$ are i.i.d. $\mathcal{N}(0,\sigma^2 / m)$. Define $$Z_m =\max_{k=1,\dots,m}X^{(m)}_k.$$...
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290 views

Determining shape parameter for Generalized Pareto Distribution Scipy

I have a set of values to which I want to fit a Generalized Pareto Distribution. Scipy provides functions for doing so: https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats....
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Estimating error for a sample minimum

Let's say I'm benchmarking some computer program and, due to non-randomness in the input data, I'm interested in the minimum running time (as opposed to the average over random inputs). In addition to ...
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Difference order distribution of extremal values of two uniform sequences

Fix an integer $m$. Pick two random integer sequences ($a_1$ to $a_n$ and $b_1$ to $b_{n'}$) uniformly independently from $[1,m]$. Denote $a(i)$ to be $i$th smallest value of $a$ and $b(j)$ to be $j$...
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CDF of maximum of correlated Gaussian variables? [duplicate]

Suppose we have a large number of correlated standard Gaussian random variables: X1, X2, .., Xn, with correlation matrix C. What is the CDF of Y=max(Xi)?
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63 views

how to deal with extreme values of longitudinal biomarker?

My dataset includes repeated measured longitudinal biomarker values on cancer patients. For example, every patient would take CEA (Carcinoembryonic antigen) test every 8 weeks. Now the problem is that ...
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EVT or directly model block maxima?

The block-maxima approach of EVT deals with the maxima of disjoint blocks of the original data. The Fisher-Tippett theorem shows that the block maximum asymptotically follows a Generalized Extreme ...
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1answer
65 views

How to construct a function with given local minima?

I need to construct a function $f(x,y)$ in which there are 3 minima: 2 local and 1 global as given below. Locals are: z = f(0.2,0.3) = 0.7 | z = f(0.6,0.8) = 0.8 Global is: z = f(0.85,0.5) = 0.6 As ...
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Consecutive differences of a uniform law

Let $N>0$ be the number of considered samples. We draw $x_1, \ldots, x_n$ from a uniform distribution over $[0;1]$. We compute $y_1, \ldots, y_{n-1}$ the differences of the sorted $(x_i)_i$. I'd ...
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89 views

Rewards in reinforcement learning for minimization problem

I am new to ML/DL/RL. I am looking to solve the classic travelling salesman problem (TSP), where the salesman has to visit all cities only once and finding the smallest path to do that (minimize ...

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