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.
Questions tagged [extreme-value-analysis]
113 questions
10
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0 answers
the parametrization of a Gumbel in terms of a Gaussian
Extreme Value Distribution From a Gaussian. I was wondering how the parametrization of $\alpha$ and $\beta$ of a Gumbel $e^{-e^{-\frac{x-\alpha }{\beta }}}$ was done in terms of a cumulative Gaussian $F(x)^n$ (where $n$ is the number of…
Nero
- 3,779
7
votes
1 answer
Extreme value theory: asymptotic of the least-rolled number out of a series of rolls
Choose positive integers $D$ and $N$. Roll a fair $D$-sided die $N$ times, recording the number of times each of the $D$ outcomes are rolled, say $r_1, r_2, \ldots, r_D.$ What are the asymptotics of $\min(r_1, r_2, \ldots, r_D)$?
They're not…
Charles
- 32,999
6
votes
1 answer
Proof of minimal eigenvalues possible?
On a German website I discovered a conjecture about a special square matrix. This conjecture contains a statement about the absolute value of the smallest eigenvalue with two formulas in respect to the number of matrix rows/columns (odd/even…
6
votes
1 answer
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, i.e. $\max(Z_1,\ldots,Z_n)$ ? (exclude the trivial…
Karl
- 704
6
votes
1 answer
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 question and hopefully address a larger audience;…
Mark
- 65
5
votes
2 answers
Record indicators in non-stationary random variables
Assume $X_1,...,X_n$ are independent, but not identically distributed continuous RVs.
I am interested in the record indicators $R_j = \mathbf{1}_{X_j > max(X_1,...,X_{j-1})}$.
According to Nevzorov (Records: Mathematical Theory) these $R_j$ are not…
AdaLovelace
- 135
5
votes
1 answer
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 sequence with common continuous distribution $F$.…
VHS
- 53
4
votes
2 answers
If sub-exponential distribution condition holds for some $n\ge2$ then it holds for all $n\ge2$
For non-negative random variables $X_1,...X_n\overset{\text{iid}}{\sim} F$, $F$ is said to be a sub-exponential distribution if
$$\lim_{x\rightarrow\infty}\frac{P(X_1+...X_n>x)}{P(X_1>x)}=n$$
for some $n\ge2$.
Result. If this holds for some value of…
zaira
- 2,386
- 2
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- 37
4
votes
1 answer
Extrema of $\sum_{j=0}^{n-1} \frac{1}{|z-a_j|^2}$ for $z$ on unit circle
Let $n \in \mathbb{N}$, $0
Beno Učakar
- 88
4
votes
1 answer
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 process into higher dimensions, we are introduced to the…
4
votes
4 answers
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:
$P_1=(x,0),x\in \mathbb R$
$P_2=(0,y),y\in \mathbb R$
$P_3=(2,3)$
First two…
Kasata Ata
- 155
3
votes
0 answers
Large deviation principle for maximum of a sequence of independent Gaussians with different variances
Let $d,n \to \infty$ with fixed $\log(n)/d \to \alpha$, for some fixed $\alpha>0$. Thus, both $n$ and $d$ are large by $n$ is exponentially larger than $d$. Let $g_1,...,g_n$ be iid from $N(0,1)$ and set $G_i := \sigma_i g_i$, for some bounded…
dohmatob
- 9,753
3
votes
0 answers
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 $Z_n$ is in the domain of attraction of the…
olappi
- 31
3
votes
1 answer
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= \max\limits_{i \in m} [\mathcal{P}(x_i \leq…
xsari3x
- 207
3
votes
1 answer
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
$1-H_\xi (x)$ approximatle equals $e^{-x}$ for $\xi=0$ for…
Florian
- 75