# 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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### 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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### 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 ...
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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### 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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### 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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### 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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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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### 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 $$Z_m =\max_{k=1,\dots,m}X^{(m)}_k.$$...
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 ...
26 views

### 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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### 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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### 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 ...
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 ...