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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22 views

A basic property of slowly varying functions

It seems that a fundamental property of a slowly varying function is that for all $\delta > 0$ $$ \lim_{x\to \infty} L(x) \, x^{-\delta} = 0.$$ How to prove this? The only book I found (that was ...
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32 views

Expectation of maxima of $n$ Erlang-distributed (Gamma-distributed) random variables for small $n$

Say I have $n$ i.i.d. random variables $\mathbf{X}_n = \{X_1,X_2,\ldots,X_n\}$, where $X_i \sim \operatorname{Erlang}(k,\theta)$ or $\Gamma(k,\theta)$. Define $$Z_n := \max \mathbf{X}_n$$ I know that ...
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1answer
18 views

A simple limit from Extreme Value Theory

I am trying to understand the basics of Extreme Value Theory with my limited mathematical abilities. Currently, I am stuck with the following limit: $$ \lim_{\xi \to 0} \exp\Big(-(1 + \xi s)^{-\frac{1}...
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19 views

Time until the next maximum?

Suppose that every year we sample $n$ random variables from a normal distribution. We want to keep track of the maximum value. On average, what is the meantime arrival between the maxima for the first ...
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1answer
31 views

If $L(y)=\min_{x\in \mathbb{R}}h(x,y)$ is well defined in $(a,b]$ then is it also well defined on $a$?

Let $h:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a continuous function such that $\lim_{x\to \pm\infty}h(x,y)=0$ for all $y$. Then we know that if $L(y)=\min_{x\in \mathbb{R}}h(x,y)$ exists, it ...
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1answer
32 views

$y'$ is being eliminated as I solve using Eulers formula what to do?

Investigate the extremals of the functional $\int((y^2 + (x^2)\cdot y'))dx$ with limits $0$ to $1$ under the conditions $y(0) = 0$, $y(1) = A$.
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1answer
27 views

Find func extremum for the given condition [closed]

Find so-called conditional extremum attained by $f(x,y,z)=x \cdot y \cdot z;$ under the following condition: $x^2+y^2+z^2=a^2$. It is thought that solution would rely on Lagrange multipliers method. ...
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65 views

Find the extrema of the absolute value of the complex function

I have a sum of a complex function, just like the following $$ f(\phi)=\sum_{n=0}^{N-1} \exp\left\{i \left[ A\phi_n - B\cos(\phi-\phi_n)\right]\right\}, \quad \phi_n=\frac{2n\pi}{N} $$ where $A$ is an ...
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1answer
102 views

Variance of max of $m$ i.i.d. random variables

I'm trying to verify if my analysis is correct or not. Suppose we have $m$ random variables $x_i$ , $i \in m$. Each $x_i \sim \mathcal{N}(0,\sigma^2)$. From extreme value theorem one can state $Y= ...
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9 views

Truncated mean of the maximum of $n$ normally distriuted variables

What is the expected value of the maximum of $n$ normally distributed variables $x_i~\boldsymbol{N}(x_0, \sigma_x^2)$, conditional on exceeding some threshold $z$? Denoting $y=max(x_1,..., x_n)$, ...
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14 views

Proving boundedness of a function using extreme value theorem

Consider a function $f(s,e)$, satisfying the following property is bounded for some finite $L_1$ and $L_2$: \begin{equation} |f(s_1,e_1)-f(s_2,e_2)|\le L_1 \rho(s_1,s_2)+ L_2\|e_1-e_2\|_2 \end{...
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2answers
125 views

How to minimize $(x-a)^2+(y-b)^2$ subject to $ \sqrt{a}+\sqrt{b}=\sqrt{2}$?

Let $x,y$ be positive real numbers satisfying $xy \le \frac{1}{4}$. I am trying to find $$ \min_{\sqrt{a}+\sqrt{b}=\sqrt{2}}(x-a)^2+(y-b)^2$$ as a function of $x$ and $y$. Is there any way to do ...
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20 views

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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0answers
26 views

Maximum value of P=perimeters

If unit square divided by 100 rectangles with equal perimeters=P, Find Maximum value of P? I just can find minimum value, it is 2/5
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1answer
127 views

Extreme values of ratios of normal random variables

the question is: Given are two independent sequences of iid normal random variables $X_i$ and $Y_i$. Form the ratios $Z_i=X_i/Y_i$. What is known about the extreme value distribution of the $Z_i$'s,...
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1answer
23 views

Slowly varying functions and the direction of convergence

Consider a slowly varying function $L(n)$ and let $\lambda > 0$. Is it true that there exists a $N \in \mathbb{N}$ such that \begin{align} \frac{L(\lambda n)}{L(n)} \geq 1 \end{align} for all $n \...
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1answer
175 views

Gumbel distribution and exponential distribution

The Gumbel distribution term in Wikipedia says: Gumbel has shown that the maximum value (or last order statistic) in a sample of a random variable following an exponential distribution approaches ...
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1answer
37 views

Extreme Value Theorem on an Unbounded Domain

Given a smooth function $f(x)\colon \mathbb{R} \to \mathbb{R} $. Suppose $$\lim_{x\to -\infty} f(x) = 0, $$ and $$ \lim_{x\to \infty} f(x) = 0. $$ Can we claim that $f$ is bounded?
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1answer
92 views

Find the extremal of $\int_a^b (y'^2+2yy'-16y^2)\, dx$

The task is to find the extremals of: $$\int_a^b (y^{\prime2}+2yy^\prime-16y^2)\, dx$$ I used the Euler-Lagrange equation and I got: $$-2y^{\prime\prime} - 32y = 0$$ I solved the differential ...
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0answers
24 views

Converting t-year probability to 1-year probability

I have a data set of peak-over-threshold values which I have fitted a generalised Pareto distribution to. From this, I wish to determine values corresponding to certain return periods in years. I have ...
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1answer
6 views

values on the boundaries of a open interval

Say we have a simple function $f(x)=x\cdot e^{-10x}$, we want to obtain the maximum value of it in an open interval $x \in (0.2, 1)$. Loosely speaking, the maximum point is at $x=0.2$ which is not ...
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1answer
32 views

Prove that if $\frac{\partial^2 f}{\partial x^2}(\textbf{a})\neq 0$ then $\textbf{a}$ cannot be extrimum of $f$

Let $f:\mathbb{R^2} \to \mathbb{R}$ in $C^2$ and harmonic. If $\frac{\partial^2 f}{\partial x^2}(\textbf{a})\neq 0$ then $\textbf{a}$ cannot be extrimum of $f$ i am not getting any idea..can someone ...
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1answer
38 views

How does multiplication by a constant affect a Gumbel random variable

Suppose $X$ is a Gumbel (Type-1 extreme value) random variable with shape and scale parameters given by ($\mu$, $\sigma$). What is the distribution of $cX$, where $c$ is a constant?
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23 views

LDP, Contraction Principle, Freidlin–Wentzell theory Dembo and Zeitouni.

I had a problem with a simple proof in Dembo and Zeitouni. Specifically Theorem 5.6.7 page 215, which is a Freidlin-Wentzell theorem. The idea of the proof is that we approximate the original SDE : ...
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2answers
40 views

Find the minimum value of the function a(x).

$$a(x)= \sqrt{x^3}+\sqrt{x^{-3}}-4(x+\frac{1}{x})$$ One of the ways I could think of was to find out the global extreme values and proceed.But as I began doing it that it takes a lot of ...
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2answers
46 views

Finding the extrema of a $f(x,y,z) = xy+yz$ with $x+y+z =1$ and $x^2+y^2+z^2 = 6$

Find the extreme values of $$f(x,y,z) =xy+yz$$ under the constraints that $$I:x+y+z =1$$ and $$II:x^2+y^2+z^2 = 6$$ by parametrisation of the curve given by the side conditions. I think that an ...
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1answer
221 views

Are min$(X_1,\ldots,X_n)$ and min$(X_1Y_1,\ldots,X_nY_n)$ independent for $n$ to infinity?

This is a question that I posted on stats.stackexchange.com but since I received no satisfying answer but still the question was upvoted by many, I want to use the oppurtunity to further extend the ...
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1answer
61 views

Tail bound for the maximum of a reduced-rank Gaussian random vector

Suppose that $X_i, i=1,...,n$ are Gaussian random variables, each with mean equal to $0$ and variance equal to $1$. However, suppose that their covariance matrix is reduced rank (assume that such a ...
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4answers
226 views

What if there are infinite stationary points?

I want to calculate extremes of certain multivariable function $f(x,y)=(6−x−y)x^2y^3$. After solving system of derivatives $f_x=0$ and $f_y=0$ I got something like this: $P_1=(x,0),x\in \mathbb R$ $...
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1answer
24 views

Complex Critical points of a Real Valued function

Assume that the function is $f:\mathbb{R}²\to\mathbb{R}³$ Exercise: Find and classify the crititcal points of the function \begin{equation*} f(x,y)=\frac{xy}{2+x^4+y^4} \end{equation*} Attempt Find $...
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1answer
34 views

Periodic Function and Extreme Values.

Can a periodic function which is strictly monotone between its extremes have any local extremum which are not global extremum? In other words, let $e(t):[0,T]\to\mathbb{R}$ be a periodic function ...
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1answer
55 views

How to find extreme values of an $f(x,y)$ function?

I need this for my semester exams, unfortunately I was absent the day this topic was "talked about". My function is the real-valued $$f(x,y)=x-xy+x^2+y^2$$, interpreted on $\mathbb{R}^2$. Single-...
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1answer
347 views

Derivation of Gumbel Distribution

The standard generalised extreme value (GEV) distribution is given by $H_{\xi}$ which is $exp(-(1+\xi x)^{-1/\xi}$ if $\xi<>0$ and $exp(-e^{-x})$ if $\xi=0$ In the lecture notes it is stated ...
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1answer
55 views

Proof of the asymptotic equivalence of the hazard rate $h(x)$ and $(\gamma x)^{-1}$.

I am trying to prove the following: Let $h(x) = \frac{f(x)}{1-F(x)}$ the hazard rate of a distribution that has PDF $f(x)$ and CDF $F(x)$ and further, be $h$ positive and differentiable on $(x_0, \...
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1answer
39 views

Extreme points of $f(x)= 2x+\cot x$

I want to find the extreme points of the function $f(x) =2x+\cot x$. I began by observing that the function is defined on $D=\mathbb{R} - \{k\pi | k\in \mathbb{Z} \} $. Now, $f'(x) =2-\frac{1}{\sin^2 ...
3
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1answer
92 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
45 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
24 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
26 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
187 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 = ...
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1answer
39 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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1answer
24 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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1answer
73 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
65 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
47 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) - \...
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1answer
87 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
58 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 $...
0
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0answers
59 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
82 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
45 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 ...