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

I want to figure out how the fraction of the biggest sample of the sum of some iid samples from an arbitrary distribution varies. For example, I have a random variable $X \sim f(x)$, and I sampled $n$ ...
Rebecca Zorichyevna's user avatar
2 votes
2 answers
77 views

What are the extremes of the function $ f\left(x,y\right)=xy\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}} $?

$ f\left(x,y\right)=xy\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}} $ ; where a,b are constants. I have tried to differentiate the equation, but I get stuck in a very hard system after this. The answer of ...
Pedro Angelo Maciel's user avatar
1 vote
1 answer
46 views

Verification of Solution for Finding Minimum and Maximum of a Function in a Given Region

I want to find the minimum and maximum values of the function $f(x, y) = \ln(1 + 9x^2 + 4y^2)$ within the region $1 \leq \frac{x^2}{4} + \frac{y^2}{9} \leq 4$. My solution steps are as follows: ...
Ayesca's user avatar
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2 answers
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Determining maximum and minimum of the function

Determine the minimum and maximum of the function $f(x,y) = xy$ in the annulus $A = \{(x,y) \mid 1 \leq x^2 + y^2 \leq 4\}$. I'm wondering if in cases like this, where we have two conditions for the ...
mia's user avatar
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1 answer
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Maximize $\surd(K_1)+2\surd(L_1)+3\surd(K_2)+4\surd(L_2)$ with constraints $L=L_1+L_2, K=K_1+K_2$

From my computations, I have found $$K_1^{*}=K/10, L_1^{*}=L/5, K_2^{*}=9K/10, L_2^{*}=4L/5, \lambda_1=\surd(5/2K), \lambda_2=\surd(5/L)$$ My Question is: How to find extreme value function $M(K, L)$. ...
Waseem Bughio's user avatar
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Statistics of differences between two largest realizations of I.I.D. normal random variables

Consider $n$ independent, normally distributed random variables $X_i$ with mean $\mu$ and variance $\sigma$. Suppose the maximum value of the realizations of each of these random variables is $\text{...
aghostinthefigures's user avatar
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1 answer
26 views

Extremize using total differential

So I was playing around with lagrange multipliers and came to following question: suppose one wants to find an extreme value of $f(x_i)$ under the condition $g(x_i)=0$. First I define $L(x_i,\lambda)\...
emilio grandinetti's user avatar
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2 answers
59 views

Limit involving logarithms and an unknown function

I need to find a function $a(t)$ which is positive such that for some $\gamma > 0$ we have: $$\lim_{{t \to \infty}}\frac1{a(t)}\left({-\frac{1}{{\ln(1-\frac{1}{{xt}})}} + \frac{1}{{\ln(1-\frac{1}{t}...
user979120's user avatar
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Fitting a general extreme value distribution on transformed data

I have a data set $X_1, ..., X_n$ and want to fit a general value distribution to this data set using R. However, when trying to do so I get an overflow. So, my idea was to instead fit a general ...
notimportant's user avatar
2 votes
4 answers
90 views

Prove: if $x^2 + y^2 < \frac{1}{2}$, then $\cos(x+y) > 0$

For an exercise about finding the minimum of a function I have to prove that $cos(x+y) > 0$ for $x^2 + y^2 < \frac{1}{2}$. Which means that $-\frac{\pi}{2}< x + y < \frac{\pi}{2}$. However,...
Cake's user avatar
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Describing the Minimum of a Brownian Motion

I'm working on characteristic functions of GBM and implementing them in option pricing in python. In deriving the cf for the minimum of a drifted BM I make this step: First, note that by implementing ...
Jord van Eldik's user avatar
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1 answer
128 views

Expected (maximum minus minimum) of Laplacian random variables

Suppose there are $n$ IID random variables denoted as $X=(X_1,\dots, X_n)$, they follow Laplace distribution with parameter $\lambda$, denoted as $Lap(\lambda)$. That is, $$f(x)=\frac{1}{2\lambda}\exp ...
white's user avatar
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0 answers
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Which claim is correct regarding the extrema of a function?

I have a function $f$ and I know that the function has only four unique extrema values which values are fA, fB, fC, and fD; moreover, I know that If condition A holds then fA is a local maximum of ...
sara96's user avatar
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Is there any hope to find extrema points of the given function $f_z (x,y)$ without using the Hessian method?

I have a three-variable function for $(x,y)\in[-\pi,\pi]$ and $z>0$ $f_z (x,y)=A(z)(\cos x+\cos y)+B(z)(\cos 2x+\cos 2y)+C(z) (\cos x \cdot\cos y),$ in which the real functions $A(z),B(z),C(z)$ can ...
MsMath's user avatar
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1 vote
1 answer
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$f(x, y, z) := xy + z^2$ on unit circle $B := \{(x, y, z) \in \mathbb{R}^3, x^2 + y^2 + z^2 \leq 1\}$

We look at $f(x, y, z) := xy + z^2$ on unit circle $B := \{(x, y, z) \in \mathbb{R}^3, x^2 + y^2 + z^2 \leq 1\}$. A) Explain why $f$ has a minimum and maximum on $B$. B) Show that $f$ has no local ...
Willem's user avatar
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Why the limiting distribution $M_n^*$ will be non-degenerate distribution?

Let $X_1,\dots, X_n$ be iid data with common distribution $F$ and $M_n:=\max\{X_1,\dots, X_n\}$. The distribution of $M_n$: $$ P(M_n\le x)=F^n(x). $$ In the textbook, it says we proceed by looking at ...
Hermi's user avatar
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1 answer
73 views

Determine the extreme values of the function $f$ with $f(x,y)=x^2y(4-x-y)$

Determine the extreme values of the function $f$ with $f(x,y)=x^2y(4-x-y)$ for the points located inside and on the triangle delimited by the lines $x=0$, $y=0$ and $x+y=6$. Is this the function I'm ...
Savantik's user avatar
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2 answers
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Why is a+b-c least here?

This part is from the well-known book What is Mathematics. Chapter VII (Maxima and Minima) - Section 5 (Steiner’s Problem) - Article 3 (A Complementary Problem). The problem: a, b, c is the distance ...
user1206899's user avatar
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2 answers
81 views

Extreme values of function with two variables $z=\frac{ax+by+c}{\sqrt{x^2+y^2+1}}$

I've been struggling with following problem: Determine extreme values of function $$ z = \frac{ax+by+c}{\sqrt{x^2+y^2+1}}, $$ where $a^2+b^2+c^2 \neq0$. Here is my approach: As we know differential ...
Alice211's user avatar
2 votes
3 answers
334 views

Find extreme values of ellipse

I have the curve equation $$ Ax^2 + By^2 + Cxy = 1 $$ which represents ellipse with center in (0, 0) and rotated some angle. How can I find max X and Y values (not semi-axes) on this ellipse? Points ...
maestro's user avatar
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1 answer
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Relative Extrema using Lagrange Multipliers - is it correct?

Here is the problem: Find an extrema of $f(x,y)=y^2+x$ with the given condition $x^2+2y^2=1$. I solved it this way: $L(x,y,\phi)=y^2+x+\lambda(x^2+2y^2-1)$ $\frac{\delta L}{\delta x}=1+2\lambda x =0$...
Kit Kolouv's user avatar
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1 answer
53 views

Points of $f(x)=x^3$ except the origin

I want to know what properties does the non-zero points in the domain of $f(x)=x^3$ have? For example, $0$ is an inflection point, since the curvature changes sign at $0$ ($f''$ changes sign near $0$)....
Mathguide's user avatar
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Does $\mathbb E X = \infty$ imply that $\mathbb P \circ X^{-1}$ belongs to Fréchet domain of attraction?

Question Let $F$ be a distribution function and $X \sim F$ a random variable such that $\mathbb E X = \infty$. That is $\mathbb E X^+ = \infty$ and $\mathbb E X^- \in \mathbb R$. Does it follow that $...
zxmkn's user avatar
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0 answers
208 views

Distribution of the argmax of a Gaussian of diagonal variance

Let $X=(X_1,\dots,X_n)\sim N(\mu,\sigma)$ be a Gaussian random vector of $d$ dimensions with mean $\mu$ and diagonal variance $\sigma^2$. Thus, both $\mu,\sigma\in\mathbb{R}^d$. Let $I=\arg\max_{i\in[...
etal's user avatar
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2 answers
72 views

Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that $f''(x)\geq a$ for all $x \in \mathbb{R}$. Show that $f$ has an absolute minimum. [closed]

Let $ f \colon \mathbb{R} \rightarrow \mathbb{R} $ twice differentiable such that $ f''(x)\geq a $ for all $ x \in \mathbb{R} $ and some $a>0$. Show that $f$ has an absolute minimum. I'm trying to ...
Curious's user avatar
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5 votes
1 answer
170 views

Showing that the distribution of record times $(\tau_k)_{k\geq 1}$ doesn't depend on the distribution, $F$, of the records $X_i$

I read that it's possible to show that the distribution of a record time sequence doesn't depend on the distribution of the record sequence itself, but how would one do this? So $(X_i)$ is an iid ...
VHS's user avatar
  • 53
1 vote
1 answer
53 views

Global extreme points of function wrt. two variables

Hi I am currently working on a problem in theoretical chemistry and am struggling a bit. While discussing Born-Oppenheimer energy functions, I was asked to find all the extreme points and saddle ...
冰淇淋's user avatar
  • 169
2 votes
3 answers
117 views

Find maximum of $\sin{x}+\sin{y}-\sin{(x+y)}+\sqrt{3}\left(\cos{x}+\cos{y}+\cos{(x+y)}\right)$

The trick is that we can't use derivatives. If we look at $\cos{x}+\cos{y}+\cos{(x+y)}$, the function cos attains its maximum of $1$ at $0$ radians, so $x=y=0$ and $x+y=0$ and the maximum of $\cos{x}+\...
 Alice Malinova's user avatar
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1 answer
208 views

$Var(\min\{X,Y\}) \leq Var(X)$?

I have this problem, and the only thing that tells me is that X and Y are random variables that have finite variance, when $\min\{X,Y\} = X$, it is true, since $Var (X) = Var (X)$, I have the problem ...
Mary's user avatar
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1 vote
1 answer
116 views

Proof of convergence of the Taylor expansion leading to the Gumbel distribution

The Taylor expansion of the CDF of the exponential function centered at $u_n$ -- with $u_n$ being an arbitrarily large value defined as $F(u_n)=1-\frac 1 n$ -- is $$F(x) = 1 - \frac 1 n e^{\alpha_n(x-...
Antoni Parellada's user avatar
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1 answer
106 views

Intermediate step in the derivation of the Gumbel distribution

On page 19 of this freely available book by Emil Gumbel (in English) the following two equations are printed: $$\Phi_n(x_n)=F^n(x_n)\tag{3.1}$$ which stands for the probability distribution that the ...
Antoni Parellada's user avatar
1 vote
0 answers
32 views

Extreme-value problem of multi-variable function

I am reading a paper in which they solve the following problem; start $$ \begin{aligned} \mathcal{W}_{\alpha_{1}, \beta_{1}, \theta} &=\frac{1}{2}\left[\cos \beta_{1} \cos \alpha_{1}+\cos \beta_{...
Dotman's user avatar
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1 vote
1 answer
69 views

Find the minimum and maximum of the function $f(x, y) = (x + y)e^{xy}$ for all $-2 \leq x + y \leq 1$

Let we have $$f(x, y) = (x + y)e^{xy}$$. We want to find minimum and maximum values of $f$ for this set of values: $$\{\ (x, y)\ : -2 \leq x + y \leq 1\ \}$$ It seems like this task can be solved with ...
Someone's user avatar
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1 vote
0 answers
55 views

The Marcum Q- function and the limit of Modified Bessel Functions

I've recently come across the following limit: $$\lim_{n \to \infty} \left[\sum_{k=1}^{\infty} \left(\frac{a}{b}\right)^k \operatorname{I}_k(ab)\right]^n$$ where $\operatorname{I}_k$ are modified ...
lbagua's user avatar
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0 votes
0 answers
104 views

How this limit prove the Frechet Distribution has Heavy Tails?

First, we start from the GEV-distribution function: Theorem 1.1.3 (Fisher and Tippett (1928), Gnedenko (1943)) The class of extreme value fistributions is $G_\gamma(ax+b)$ with $a>0$, $b$ real, ...
Jelmer's user avatar
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1 vote
1 answer
139 views

Question on Block Maxima Method for Extreme Value Theory

I am reading about extreme value theory. Let $X_1, X_2, ...$ be i.i.d. random variables and $M_n = \text{max}\{X_1,...,X_n\}$ as usual. I understand that, by the Fisher-Tippett-Gnedenko theorem: If ...
AdaLovelace's user avatar
1 vote
0 answers
120 views

Linear mean excess function implies a generalized pareto distribution

I am trying to do the following prove: Let $X$ be a non-negative random variable. Show that if for some $a \in (-1,\infty), b>0$, it holds that the mean excess function $\epsilon_x(t)$ is: $$\...
flb's user avatar
  • 21
0 votes
2 answers
102 views

Decompose a positive number into the sum of 5 numbers in such a way that the sum of its inverse is maximal. [closed]

I have to decompose a positive number into the sum of 5 numbers in such a way that the sum of its inverse is maximal. I understand that is optimization, but I don't know how to do it with five ...
user895583's user avatar
2 votes
0 answers
43 views

Expectation of maximum among N independent but not identical distributed guassian random variables

Suppose here are n independent gaussian random variables, $X_1,X_2\dots X_n$, such that $X_i\sim \mathcal{N}(\mu,\sigma_i^2)$, let $X^*=max(X_1,X_2\dots X_n)$. If $\sigma_1=\sigma_2=\dots\sigma_n$, I ...
MissCrystal's user avatar
0 votes
1 answer
26 views

How to compute an integral with respect to $K(t, dk)$

Let $X$ be a random variable with distribution $F$ and $K: \mathbb{R}^+\times\mathbb{N}\to[0,1]$ defined by $K(t, \{k\}):=F(t)^{k-1}(1-F(t))$. If $u:\mathbb{R}^+\times\mathbb{N}\to\mathbb{R}^+$, how ...
The Mad Scientist's user avatar
0 votes
1 answer
41 views

Let $x_0$ be an extreme point of $f$ and $g$. Is $x_0$ an extreme point of $\max\{f,g\}$?

Let $x_0$ be an local extreme point of $f$ and $g$. Is $x_0$ an local extreme point of $\max\{f,g\}$? Let $x_0$ is the local maximum point of $f$ and $g$, it is OK! What may happen if $x_0$ is the ...
xldd's user avatar
  • 3,475
2 votes
2 answers
64 views

Find RV distribution and prove its convergence to zero

Let $X_n$ be a sequence of independent and uniformly distributed over $[0; 1]$ random variables. The task is to find distribution of the following random variable: $m_n=\min\{X_1, X_2, \ldots, X_n\}$ ...
Gorhonm's user avatar
  • 169
1 vote
1 answer
136 views

Properties, bounds and limits about difference of two inverse standard normal CDF variables and extreme value distribution

I'm interested in the variable: $$\sigma_n=\Phi^{-1}\left(1-{1\over n}e^{-1}\right)-\Phi^{-1}\left(1-{1\over n}\right),$$ where $\Phi(\cdot)$ is the CDF of standard normal distribution. I want to ...
Juntao Wang's user avatar
4 votes
1 answer
154 views

What is the meaning of writing the differential inside of a function?

I am reading through Resnick's "Extreme Values, Regular Variation and Point Processes" and have come across some notation that I am not familiar with. In talking about moving a Poisson point ...
The Mad Scientist's user avatar
0 votes
1 answer
39 views

Determining the local extreme values of a function

I am currently studying analysis and I am facing a problem with understanding how to determine the local extreme values of a function. I just started studying this, but our prof is already giving us ...
asprog's user avatar
  • 113
0 votes
1 answer
68 views

Why is it necessary to find extreme values of L-functions?

Apparently I have been exploring Riemann Zeta Function and have lately come across $L$-functions. After going through few papers, I realised that many mathematicians are giving their full time knowing ...
user511110's user avatar
0 votes
1 answer
22 views

What single estimator of a 5000 series of 20 elements for a non-normal distribution should be used?

The data are traffic counts of one minute (20 different days) of 5000 streets. The problem is that the mean of the 20x5000 observations does not explain much as it is not a normal distribution. Also, ...
Daniel Cuende's user avatar
2 votes
1 answer
153 views

Construction of a $C^1$ function, which is (locally) larger than a given continous function

Let $u: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continous function which is differentiable at the point $x_o \in \mathbb{R^n}$. Find a function $v \in C^1(\mathbb{R^n})$ such that $v(x_o)=u(x_o)$, $\...
XPenguen's user avatar
  • 2,281
1 vote
1 answer
57 views

Is $(0,0,0)$ saddle point?

$f(x,y,z)=x^3+y^3+z^3-3xy-3yz$ Is $(0,0,0)$ saddle point? $\nabla f(0,0,0)=0$, so $(0,0,0)$ is one of the stationary point. Also, because of my posture The reason why $f(0,0,0)$ is not a extreme value....
daㅤ's user avatar
  • 3,254
1 vote
0 answers
52 views

The reason why $f(0,0,0)$ is not a extreme value.

$f(x,y,z)=x^3+y^3+z^3-3xy-3yz$ According to wolfram alpha, $f(0,0,0)$ is not a extreme value. I considered why $f(0,0,0)$ is not a extreme value. Is this reason correct? When $x$ is very close to $0$ ...
daㅤ's user avatar
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