## FANDOM

33.145 Seiten

 Typus Verschleierung Bearbeiter Hindemith Gesichtet
Untersuchte Arbeit:
Seite: 16, Zeilen: 1-19
Quelle: Lindskog 2004
Seite(n): 10, 11, Zeilen: -
EXAMPLE 2.2.9. (Maxima of Cauchy random variables). Let X be a sequence of iid standard Cauchy random variables, i.e. the density function is given by ${\scriptstyle f(x) = \frac{1}{\pi(1+x^2)} }$ for ${\scriptstyle x \in \mathbb{R} }$. Using l’Hospitals rule we obtain
${\scriptstyle \lim_{x\to\infty}\frac{\bar{F}(x)}{(\pi x)^{-1} }= \lim_{x\to\infty}\frac{f(x)}{\pi^{-1} x^{-2} } = }$
${\scriptstyle \lim_{x\to\infty}\frac{\pi x^2}{\pi (1+ x^2) }=1 }$

Then ${\scriptstyle \bar{F}(x)\sim(\pi x)^{-1} }$ as ${\scriptstyle x\to\infty}$. Hence, for ${\scriptstyle x>0, \quad\bar{F}(nx/\pi)\sim (nx)^{-1} }$ as ${\scriptstyle n\to\infty}$ and

${\scriptstyle \mathbb{P} (M_n\le nx/\pi ) = \left(1- \bar{F} (nx/\pi)\right)^n }$
${\scriptstyle = \left( 1-\frac{1}{n}\left(\frac{1}{x} + o(1)\right)\right)^n }$
${\scriptstyle \to \quad \exp\left\{ -x^{-1}\right\}= \Phi_1 (x), }$

for ${\scriptstyle x>0}$. Hence F belongs to the maximum domain of attraction of the Frèchet [sic] distribution. The normalizing constants can be chosen to be ${\scriptstyle a_n=n }$ and ${\scriptstyle b_n=0 }$

EXAMPLE 2.2.10. (Maxima of exponential random variables). Let X be a sequence of iid standard exponential random variables, i.e., the distribution function F of X is given by ${\scriptstyle F(x) = 1- e^{-x} }$ for ${\scriptstyle x\ge 0 }$ Then,

${\scriptstyle \mathbb{P} (M_n - \ln n\le x ) = F( x+\ln n)^n }$
${\scriptstyle = (1-\exp(-x-\ln n ))^n }$
${\scriptstyle = (1-n^{-1} e^{-x} )^n }$
${\scriptstyle \to \quad \exp ( -e^x )= \Lambda (x) , \quad x\in\mathbb{R} }$

Hence, F belongs to the maximum domain of attraction of the Gumbel distribution. The normalizing constants can be chosen to be ${\scriptstyle a_n=1 }$ and ${\scriptstyle b_n= \ln n }$

Example 3.2 (Maxima of exponential random variables). Let (Xk) be a sequence of iid standard exponential random variables, i.e. the distribution function F of Xk is given by${\scriptstyle F(x) = 1- e^{-x} }$ for ${\scriptstyle x\ge 0 }$ Then
${\scriptstyle \mathbb{P} (M_n - \ln n\le x ) = F( x+\ln n)^n }$
${\scriptstyle = (1-\exp\{-x-\ln n\})^n }$
${\scriptstyle = (1-n^{-1} e^{-x} )^n }$
${\scriptstyle \to \quad \exp \{-e^{-x}\} = \Lambda (x) , \quad x\in\mathbb{R} }$

Hence, F belongs to the maximum domain of attraction of the Gumbel distribution ${\scriptstyle (F \in MDA(\Lambda))}$. The normalizing constants can be chosen to be ${\scriptstyle a_n=1 }$ and ${\scriptstyle b_n= \ln n }$.

Example 3.3 (Maxima of Cauchy random variables). Let (Xk) be a sequence of iid standard Cauchy random variables, i.e. the density function f of Xk is given by ${\scriptstyle f(x) = (\pi(1+x^2))^{-1} }$ for ${\scriptstyle x \in \mathbb{R} }$. Using l’Hospitals rule we obtain

${\scriptstyle \lim_{x\to\infty}\frac{\bar{F}(x)}{(\pi x)^{-1} }= \lim_{x\to\infty}\frac{f(x)}{\pi^{-1} x^{-2} } = }$
${\scriptstyle \lim_{x\to\infty}\frac{\pi x^2}{\pi (1+ x^2) }=1 }$

i.e. ${\scriptstyle \bar{F}(x)\sim(\pi x)^{-1} }$ as ${\scriptstyle x\to\infty}$. Hence, for ${\scriptstyle x>0, \quad\bar{F}(nx/\pi)\sim (nx)^{-1} }$ as ${\scriptstyle n\to\infty}$ and

${\scriptstyle \mathbb{P} (M_n\le \frac{nx}{\pi}) = \left( 1- \bar{F}\left(\frac{nx}{\pi}\right)\right)^n }$
${\scriptstyle = \left( 1-\frac{1}{n}\left(\frac{1}{x} + o(1)\right)\right)^n }$
${\scriptstyle \to \quad \exp\left\{ -x^{-1}\right\}= \Phi_1 (x), \quad x > 0. }$

[Seite 11]

For ${\scriptstyle x\le 0, \mathbb{P}(M_n\le nx/\pi )\le F (0)^n \to 0 }$. Hence, F belongs to the maximum domain of attraction of the Fréchet distribution ${\scriptstyle (F \in MDA(\Phi_1))}$. The normalizing constants can be chosen to be ${\scriptstyle a_n=n/\pi }$ and ${\scriptstyle b_n=0 }$.

 Anmerkungen Ausser der Reihenfolge der Beispiele sind die Unterschiede zwischen Quelle und Dissertation sehr gering: es gibt in der Dissertation geringfügige Auslassungen und eine fehlerhafte Normalisierungskonstante. Es mag sich hier zwar um Standardbeispiele handeln, aber der Wortlaut und die mathematische Herleitung Term für Term sind übernommen ohne dass die Quelle genannt wird. Sichter (Hindemith), WiseWoman