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Questions tagged [extreme-value-analysis]

This tag is for questions pertaining to the probabilistic/statistical theory of extreme deviations from the median of probability distributions. A central result of this theory is the Fisher–Tippett–Gnedenko theorem. It is not to be confused with the extreme-value-theorem tag that refers to a theorem for real valued continuous functions on a closed and bounded interval.

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38 views

For which values of $\alpha \geq 0$ is the function $\exp(\log^\alpha(x))$ varying regularly?

I am trying to prove for which values of $\alpha \geq 0$ the following function $f(x) := \exp(\log^\alpha(x))$ is varying regularly (or slowly) and if it is varying, I need to determine the index of ...
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0answers
8 views

How to determine if a critical point is a min/max on a function $f(x,y)$ on MATLAB

I am supposed, given a critical point [x,y] on a function f(x,y), determine if the point is a max/min without using the second derivative test. So I have a program which calculates crtical points. ...
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0answers
30 views

Limit distribution for minimum of n random draw from non-central chi square

Suppose a random variable 𝑋 follows a non-central chi square distribution.Suppose we take n samples from this distribution and only record the minimum value. $$\ M_n=min(X_1,X_2,...,X_n) \ \ \ \ \ \ ...
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0answers
25 views

Mean excess function for Pareto-type distributions

I have to compute the MEF for Pareto-type distributions, that is distributions $F_{X}$ such that $$1-F_{X}(x)=x^{-1 / \gamma} \ell_{F(x)} \quad \gamma >0,$$ where $\ell_{F(x)}$ is a slowly varying ...
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0answers
48 views

Question about Fisher-Tippett-Gnedenko theorem

I'm studying about Fisher-Tippett-Gnedenko theorem. In web I see this representation of the theorem"Jack D'Aurizio (https://math.stackexchange.com/users/44121/jack-daurizio), Expected value of minimum ...
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0answers
19 views

Local maxima on multivariable function using rects

Lets suppose you have a function $ f:\mathbb R^2 \rightarrow \mathbb R $ and you wanna know if a point $(x_0,y_0)$ is a local maximum. If you equate $y=m(x-x_0)+y_0$, then analyze the resulting ...
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0answers
24 views

$f(x)=\frac{x^4}{4}+\frac{x^3}{3}-x^2$. Find the intervals on which $f$ is increasing and decreasing, and identify the local extrema of $f$.

Let $f:\Bbb{R} \to \Bbb{R}$ be a function defined by $f(x)=\frac{x^4}{4}+\frac{x^3}{3}-x^2$. Find the intervals on which $f$ is increasing and decreasing, and identify the local extrema of $f$. My ...
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2answers
103 views

Asymptotics of the minimum of Binomial random variables

Let $Y=\min\{X_1, X_2 \cdots X_k\}$ be the minimum of $k$ iid Binomial $(n,1/2)$ random variables. I'm interested in the asymptotics of $Y$ (distribution, or mean and variance) for large $n$ and $k = ...
0
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1answer
30 views

Check that $x$ can be taken to be an extremal point of $A$

Let $T : \Bbb R^n \to \Bbb R$ be a linear map. Let $A \subset \Bbb R^n$ be a convex, closed and bounded set. We know that there exists $x_0 \in A$ such that $\sup T(x) = T(x_0)$. Check that $\...
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0answers
19 views

Taylor series expansion of quantile function

Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $. We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$. Do you have any ...
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1answer
19 views

Extreme values of function

We have $A = \overline B(O_3, 1)$\ {$O_3$} and $f:A \to \mathbb R$, $f(x,y,z) = \frac{x+y+z}{\sqrt{x^2+y^2+z^2}}$. The problem asks for extreme values of function $f$ and asks if $f$ is touches them.
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0answers
35 views

Stationary values of ${x^3 + y^3 + z^3 - 3xyz}$ when ${ax + by + cz =1}$

If ${ax + by + cz =1}$, then show that in general ${x^3 + y^3 + z^3 - 3xyz}$ has two stationary values ${0}$ and $\frac{1}{(a^3+b^3+c^3-3abc)}$, of which first is max or min according as ${a+b+c>0}$...
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0answers
12 views

Help check my derivation of multivariable functions extremum criterion

I'm trying myself to derive the criterion of extremum of several variables. Let $f:\Bbb R^2\to\Bbb R$, $c=(x_0,y_0)\in\Bbb R^2,~u=(a,b)\in\Bbb R^2$ be a unit vector. Define $r(t)=(x_0+at,y_0+bt)$. ...
0
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1answer
57 views

Proof of AM GM theorem using Lagrangian

Given: $\prod_{i=1}^n x_i = 1$ leads to constraint function $G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$ ($\prod_{i=1}^n x_i =x_1 x_2...x_n$) Task is to to find the minimum using conditional extrema of ...
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0answers
43 views

Inequality with extreme values

$f:[0,1]\rightarrow \mathbb{R}$ is continuous and non-constant function. $F$ is the indefinite integral of $f$ such that $F(0)=F(1)=0$. $m$ and $M$ are the minimum and the maximum values of $f$. Now ...
2
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1answer
35 views

Absolute conditional minimum of function in n-dimensional space

Function $$F(x_1,x_2,...,x_n) = \sum_{i=1}^n x_i$$ on the constraint $$G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$$ what I come up with is writing down the Lagrangian: $$L = F(x_1,x_2,...,x_n) - \...
1
vote
1answer
77 views

AM GM inequality using conditional extrema

Given: $\prod_{i=1}^n x_i = 1$ leads to constraint function $G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$ ($\prod_{i=1}^n x_i =x_1 x_2...x_n$) Task is to to find the minimum using conditional extrema of ...
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2answers
48 views

Show median zero and symmetry of differences of Gumbel [closed]

Consider a random vector $\epsilon\equiv (\epsilon_1, \epsilon_2, \epsilon_0)$. Suppose $\epsilon_1, \epsilon_2, \epsilon_0$ are i.i.d., distributed as Gumbel with location $0$ and scale $1$. Take $...
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0answers
27 views

Problem with finding extreme values with TI84

I'm trying to find the extreme values of a function with TI84 plus CE. I have found the same instructions on several sites and in the manual. However, I don't get the correct resuluts. Example: f: ...
2
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2answers
56 views

Extreme Value Problem: Minimizing vs. Maximizing

Problem: Determine the distance between $(0,0,0)$ and the straight line which we get by intersecting the two planes $x-y+z=1$ and $x+y-z=0$. Distance between a point and origin: $f(x,y,z)=\sqrt{x^2+y^...
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1answer
39 views

How to find extrema for given implicit function of $(x,y,z)$ variables?

Given the equation $$z^3 - 33xyz - 27 = 0,$$ we know that $z=z(x,y)$. As $z$ is function of both $x$ and $y$, I don't really know how to search for extremeum, because of three variables. I only ...
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1answer
32 views

Finding global Max/Min in multi valued function with boundaries

so this is an exmaple from a book I have. Find Max/Min of $f=\frac{x^2}{2} + \frac{y^2}{2}$ on $E=\big\{ (x,y)\in \mathbb R \big| \frac{x^2}{2} + y^2 \leq 1\big\}$ 1. Step: We look at the Interior ...
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0answers
19 views

generalized extreme value for independent but non-identical sequences?

The generalized extreme value (GEV) distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. I am ...
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0answers
33 views

Asymptotic result of $\mathbb{E}\left[\max_{i=1,\cdots,K} |h_i|^2 \right]$?

I am trying to understand the following asymptotic results in one of the article which gives no proof (and no reference either). I just can't see how trivial this is. Given i.i.d. $h_i \sim \mathcal{...
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1answer
52 views

What is the extreme value distribution for the Kolmogorov-Smirnov D statistic?

I occasionally find that I want to apply a K-S test in the context of unit-testing software that involves random behaviors. Unit testing with sampling statistics is a bit tricky because you want to ...
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0answers
22 views

Fraction of the largest element of a sum of $N$ i.i.d. random variates sampled from power law distribution

For a probability distribution $$ p(x) \propto x^{-(\mu + 1)} \qquad 0 < \mu < 1$$ both the sum of $N$ i.i.d. samples $S_N$ and the largest element of those samples $x_{\text{max}}$ scale as $N^{...
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1answer
128 views

Generalized Pareto distribution (GPD)

I'm trying to understand the functional form of the Generalized Pareto distribution (GPD) presented in Wikipedia. In the "Definition" section location parameter $\mu$ does not appear in the function, ...
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0answers
54 views

EVT: correlation standard normal distribution and Gumbel distribution

The standard normal distribution given here by $F$, satisfies the following limit: $$ n(1-F(a_n x+b_n)) \rightarrow e^{-x} \;\;\text{ as }\;\; n \rightarrow \infty$$ with $$b_n=(2\log(n)+\log(\...
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0answers
51 views

Block size BMM when modeling Generalized Extreme Value distribution

I'm an Operations Research student who's trying to wrap his mind around extreme value theory. I've read into EVT and more specific into the first theorem of Fisher, Tippett and Gnedenko. In their ...
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0answers
192 views

Behaviour of the solution of $f'(x) = \beta f(x / \alpha)$ with $0 < \alpha < 1$

I have a function $f(x)$ defined for $x \in [0,\, 1]$ which is such that $$ f'(x) = \beta \,f(x /\alpha) \qquad\text{for all } x \text{ with } 0 \leq x \leq \alpha, $$ where $\beta$ and $\...