Questions tagged [extreme-value-theorem]

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Proof Verification for Extreme Value Theorem

I had an idea for a proof of the Extreme Value Theorem, and I was wondering if it was valid. Any advice you have would be greatly appreciated. Thank you! Prove: If a function f(x) is continuous on a ...
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How to fully estimate a probability density from only a sample of minimum values?

We are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$. By means of ...
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Open sets in a particular norm.

Let $f \in C^2[0,1]$, define $||f||_{2,\infty} := \underset{x \in [0,1]} {\text{sup}} |f(x)|+ \underset{x \in [0,1]} {\text{sup}}|f'(x)|+ \underset{x \in [0,1]} {\text{sup}}|f''(x)|$. Then I wish to ...
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Asymptotic behavior of the maximum of independent but non-identically distributed Gaussian random variables

Is there any result on the asymptotic behavior of the maximum of independent but non-identically distributed Gaussian random variables? Something similar to the result by Gnedenko, (1947) that for a ...
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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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Inproper Extreme Value Distribution of Type I (Gumbel) or badly estimated parameters?

I would appreciate a lead\verification on the following excercise. Let $W$ denote the anual maximum wind load. The anual maximum of $W$ is distributed according to an asymptotic extreme value ...
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Find a max possible value of an undefined function in an interval when derivative is a constant [closed]

The function $f(x)$ is continuous and differentiable in $[0,1]$ if $f'(x)\le 10$ for all $x\in[0,1]$ and $f(0)=0$, What is the maximum possible value of $f(x)$ for $x\in [0,1]$ ? Any help would be ...
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Show that if f is continuous and periodic then f attains both its minimum and its maximum.

Show that if f is continuous and periodic then f attains both its minimum and its maximum. The solution is given below: But I wonder why he choose the k like this and is the solution does not ...
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Extreme value theory - proof this is a poisson point process

Let $(X_n)_{n \geq 1}$ be an i.i.d sequence of real valued RVs with continous distribution function f and $M_n:=\max \{X_1,...,X_n \}$. Let $U_n:=\inf \{ k \in \mathbb{N} | X_k>X_{U_{n-1}} \}$ be ...
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Proving there must exist a maximum value in a continuous interval

I know this is part of the Extreme Value Theorem but I want to tackle this one bit first, focusing on the maximum case. For a function $f$ continuous over interval $[a,b]$, it has a max value over ...
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, ...