Questions tagged [extreme-value-theorem]

This tag is for questions relating to "Extreme Value Theorem". The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions.

24 questions with no upvoted or accepted answers
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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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17 views

Tail of CDF of non-central chi squared RV using asymptotics of Bessel function

The pdf and cdf of the non-central chi squared RV (under the scenario I am studying) is given as follows: \begin{align} &f(x)=\frac{1}{v} \exp\left(\frac{-(a+x)}{v}\right)I_{0}\left(\frac{\sqrt{xa}...
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2answers
70 views

Can the ratio of the two smallest element of an iid sample converge to 1 if the support of $X$ is positive?

We have: $\mathbb P(X \leq 0)=0$ and $\mathbb P (X \leq a)>0$ for any $a>0$.
1
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1answer
53 views

Second derivative test in the Hilbert space case

Let $H,E$ be $\mathbb R$-Hilbert spaces; $f\in C^2(\Omega)$; $c\in C^2(\Omega,E)$; $M:=\left\{c=0\right\}$; $x\in M$ be a local minimum of $f$ constrained on $M$, i.e. $$f(x)\le f(y)\;\;\;\text{for ...
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2answers
34 views

Example of a calculus optimization problem where the answer occurs at an endpoint

I'm teaching optimization problems in calculus right now. An easy example would be something like: Find the dimensions of a rectangle with perimeter $100$ m whose area is as large as possible. The ...
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1answer
22 views

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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56 views

Domain of attraction $F(x)=\exp(-x-\sin(x))$

I need to show that $F(x)=\exp(-x-\sin(x)),x >0$ is in no domain of attraction. That means that there exist no $a_{k} >0, b_{k} \in \mathbb{R}, k \in \mathbb{N}$ with $\lim\limits_{k \to \infty}...
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32 views

Critical point of a multivariable function

For the function $f(x,y)=x^2y^2 + \frac{1}{y} + \ln(\frac{1}{x})$ I get two critical points, namely $P_1 =\left(\sqrt{\frac{1}{2}},1 \right)$ and $P_2 =\left(-\sqrt{\frac{1}{2}},1 \right)$. However ...
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34 views

Rate of convergence of the maxium of a random sequence

I came across a problem which requires the rate of convergence of $\sup_{1\le j \le N} |\sum_{i=1}^{N}X_{i,j}/N|$. If sup over a finite number of objects, this is a simple application of LLN. However, ...
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89 views

Frechet distribution function compared to empirical data

So basically I am trying to evaluate VaR on the Tesla stock using a Block Maxima method. I.e. I assume that the worst weekly log returns follow a generalized extreme value distrubtion with the shape ...
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11 views

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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36 views

If $u$ is an extremal point of a functional $E$ and $γ$ is a smooth curve with $γ(0)=u$, then $0$ is an extremal point of $E\circ\gamma$

Let $X$ be a $\mathbb R$-Banach space, $M\subseteq X$ be open, $M_0\subseteq M$ be closed, $E\in C^1(M)$, $u\in M_0$ be an extremal point of $E$, $\varepsilon_0>0$ and $\gamma\in C^1(-\varepsilon_0,...
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2answers
43 views

Probability that a normal random variable is greater than the maximum of $n$ i.i.d. random variables

Let $M_n = \max\{X_1,\ldots,X_n\}$, where $X_1,\ldots,X_n$ are i.i.d. random variables. We know about $M_{n}$ from the extreme value theorem and know it's mean and variance. Let $Y \sim \mathcal{N}(\...
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33 views

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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23 views

Higher order derivative and extrema, theorem.

I wanted ask, maybe someone here can give a site or other source as to where could I find a theorem about higher order derivatives. The conclusion of theorem is as follows: If $f'(x_0)=f''(x_0)=f'''(...
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16 views

Extreme value theory Insurance

I've got data-set which is very large (16 million + ) and has over 150 covariates (which some are hot-encoded). This data-set spans across roughly 10 years These covariate inlcude details of the ...
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1answer
26 views

Finding points which are not local maximum or minimum

Consider the picture below: This is the levels curves for a function $f(x,y)$ where: Blue line is the partial derivative of $f(x,y)$ with respect to x Red line is the partial derivative of $f(x,y)$ ...
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10 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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20 views

Creating variable functions using MATLAB

So I have three seperate function in MATLAB where each have its designated purpose. The first one calculates the partial derivative The second finds the roots for a system of two equations and two ...
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1answer
52 views

Showing that the pointwise limit of continuous functions equals its supremum somewhere on compact domain

I'd appreciate hints on proving the following theorem: If $f(x) = \lim_{n\to\infty} f_n(x)$ for each $x \in [0,1]$ and $M = \sup_{x\in[0,1]} f(x)$, then there is $t \in [0,1]$ such that $f(t) = M$. ...
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17 views

Reverse Hoeffding Inequalities

Suppose that $X_t$ is a super-martingale, the Hoeffding inquality gives an exponential upper bound on the quatity $$ \mathbb{P}\left( \sup_{0 \leq t\leq T}X_t \geq x \right). $$ When can a lower-...
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21 views

Show that a point is closest to another point with extreme value theorem

Show there is a point of the plane $\{x \in \mathbb{R^3} \mid x_1 + 2x_2 + 3x_3 = 13\} = A$ closest to the point $(1, 1, 1)$. Let $B[(1, 1, 1), r]$. Let a function $f: A\cap B \rightarrow \mathbb{R}$...
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1answer
38 views

Proving the maximum value of a set of numbers given their sum

Say we have $\sum_{i=1}^n x_n = C$, i.e., $x_1+x_2+...x_n=C$ where C is a constant and $x_1,x_2,...x_n$ are nonnegative. Prove the product $(x_1)(x_2)(x_3)...(x_n)$ has a maximum if and only if $x_1=...
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1answer
577 views

What are the points on a Return Level Plot?

A pretty simple question I think, but I can't seem to find an answer anywhere. I have produced the below return level plots for a project I am completing on stock market data. I understand the meaning ...