Questions tagged [extreme-value-analysis]

Question 1

Let $X \perp X' \sim \cal{N}( 0, \Sigma )$ and define $\bar{X}_{i \leftrightarrow j} := \frac{1}{\sqrt{2}} X + \frac{1}{\sqrt{2}} P_{i \leftrightarrow j} X'$ with $P_{i \leftrightarrow j}$ the (...

Question 2

With $Z^{(n)} \in \mathbb{R}^n$ a random vector of $n$ i.i.d. samples from the standard normal distribution and $\max Z^{(n)} \equiv \max_i Z^{(n)}_i$, $\mathbb{E}[ \max Z^{(n)} ] / \sqrt{ n - 1 }$ ...

Question 3

Consider the function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x,y) = x^4 + y^3$. We have The only stationary point is at $(0,0)$, where $f(0,0)=0$. The Hessian matrix evaluated at $(0,0)$ is $$...

Question 4

Consider a sequence of random variables $(X_n)_{n\in \mathbb{N}}$ and a sequence of positive reals $(\sigma_n)_{n\in \mathbb{N}}$ such that: $$\frac{X_n}{\sigma_n}\stackrel{d}{\longrightarrow}Fréchet(\...

Question 5

Suppose I have a centered stationary Gaussian process $X_t, t\in[0,1]$ for which $X_0 = -X_1$ a.s. I am interested in the lower bound for the tail probability $P(\sup_{t\in(0,1)}|X_t| > x)$ for ...

Question 6

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 ...

Question 7

Suppose we have $n$ random variables, $X_1, \cdots ,X_n$ each one with a binomial distribution with the same parameter $N$ but different probability $p_j$. $X_j \sim Binom(N,p_j)$ for $1\leq j \leq n$ ...

Question 8

Let $X$ be a random variable such that for all $x>0$, some $t_0>0$, some $\sigma(t_0)>0$, and some $\gamma \in \mathbb{R}$, $$\frac{P(X>x+t_0)}{P(X>t_0)} = (1+\gamma \frac{x}{\sigma(t_0)...

Question 9

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 $N(0,1)$ ...

Question 10

For a sub-exponential distribution $F$, we know that for $X_1, \ldots, X_n\overset{\text{iid}}{\sim}F$ and any $k=1,2,\ldots,n$: $$\lim_{x\rightarrow\infty}P(X_k>x|X_1+X_2+\ldots+X_n>x)=\frac{1}{...

Question 11

For non-negative random variables $X_1,...X_n\overset{\text{iid}}{\sim} F$, $F$ is said to be a sub-exponential distribution if $$\lim_{x\rightarrow\infty}\frac{P(X_1+...X_n>x)}{P(X_1>x)}=n$$ ...

Question 12

In Theorem 1.1.8 of the book Extreme value theory. An introduction, the conditions are stated as follows: Let $F$ be a distribution function and $x^* = \sup \{x \colon F(x)<1\}$ its right endpoint. ...

Question 13

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 ...

Question 14

We have $\left\{\begin{matrix} xyz \rightarrow extr\\ \frac{1}{2}x^2+\frac{1}{2}y^2+\frac{1}{2}z^2=1 \\ x+y+z=0\end{matrix}\right.$ Let's find Lagrange Multipliers: $L = \lambda_0(xyz) + \lambda_1(\...

Question 15

Let $f = (x^2+y^2) \exp(-x^2-y^2)$. Find global extremums. Put $u = \exp(-x^2-y^2)$ $f_x = 2xu - 2xu(x^2+y^2) = 0$ $f_y = 2y*u - 2y*u*(x^2+y^2) = 0$ As $u \neq 0$ we have $2x(1 - x^2 - y^2) = 0 \...

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Questions tagged [extreme-value-analysis]

Schur concavity of $\mathbb{E}[ \max X ]$, $X \sim \cal{N}( 0, \Sigma )$, as a functional of $\Sigma$?

Monotonicity of $\mathbb{E}[ \max Z^{(n)} ] / \sqrt{ n - 1 }$

Stationary point not a local extremum nor a saddle point

Asymptotic integrability of truncation by arbitrary constant

Bound on the tail probability of the supremum of the absolute value of a Gaussian process

Proof of minimal eigenvalues possible?

Distribution of the max of independent non-identically distributed binomial random variables

Mean Residual life plot proof(extreme value theory)

Large deviation principle for maximum of a sequence of independent Gaussians with different variances

Tail asymptotic behaviour of normal distribution

If sub-exponential distribution condition holds for some $n\ge2$ then it holds for all $n\ge2$

Theorem 1.1.8 in Extreme value theory. An introduction: Condition on density and right endpoint

Record indicators in non-stationary random variables

Find global extremums of function

Show local extr of function is global one.

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