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.
116 questions
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Stationary point not a local extremum nor a saddle point
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
$$...
- 2,712
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Asymptotic integrability of truncation by arbitrary constant
Consider a sequence of random variables $(X_n)_{n\in \mathbb{N}}$ and a sequence of positive reals $(\sigma_n)_{n\in \mathbb{N}}$ such that:
$$\frac{X_n}{\sigma_n}\stackrel{d}{\longrightarrow}Fréchet(\...
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71
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Bound on the tail probability of the supremum of the absolute value of a Gaussian process
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 ...
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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 ...
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Distribution of the max of independent non-identically distributed binomial random variables
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 $1\leq j \leq n$
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- 183
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Mean Residual life plot proof(extreme value theory)
Let $X$ be a random variable such that for all $x>0$, some $t_0>0$, some $\sigma(t_0)>0$, and some $\gamma \in \mathbb{R}$,
$$\frac{P(X>x+t_0)}{P(X>t_0)} = (1+\gamma \frac{x}{\sigma(t_0)...
- 2,316
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Large deviation principle for maximum of a sequence of independent Gaussians with different variances
Let $d,n \to \infty$ with fixed $\log(n)/d \to \alpha$, for some fixed $\alpha>0$. Thus, both $n$ and $d$ are large by $n$ is exponentially larger than $d$. Let $g_1,...,g_n$ be iid from $N(0,1)$ ...
- 9,803
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Tail asymptotic behaviour of normal distribution
For a sub-exponential distribution $F$, we know that for $X_1, \ldots, X_n\overset{\text{iid}}{\sim}F$ and any $k=1,2,\ldots,n$:
$$\lim_{x\rightarrow\infty}P(X_k>x|X_1+X_2+\ldots+X_n>x)=\frac{1}{...
- 2,396
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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
$$\lim_{x\rightarrow\infty}\frac{P(X_1+...X_n>x)}{P(X_1>x)}=n$$
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- 2,396
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Theorem 1.1.8 in Extreme value theory. An introduction: Condition on density and right endpoint
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. ...
- 2,316
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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 ...
- 135
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Find global extremums of function
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(\...
- 85
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Show local extr of function is global one.
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 \...
- 85
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Conditional Distribution of $X - u$ Given $X > u$ for a GPD
I am asked to determine the conditional distribution of $X - u$ given $X > u$. Given that $X$ follows the Generalized Pareto Distribution (GPD) with the cumulative distribution function $H(x; \...
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Maximum of $M$ random variables which are the maximum of $m$ normal distributed variables.
Let $X_{i,1}, \dots, X_{i, m}$ be a collection of normally distributed random variables $\mathcal{N}(0,1)$, and let $X_{i, (m)} = \max_{j\leq m}X_{i, j}$. I know from extreme value theory that $X_{i, ...
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