Adakah contoh variabel bebas (kasar) yang bergantung pada nilai ekstrim?

Saya mencari contoh 2 variabel acak $X$ , $Y$ such that

| c o r (X, Y) | \approx 0

$\newcommand{\cor}{{\rm cor}}|\cor(X,Y)| \approx 0$

but when consider the tail part of the distributions, they are highly correlated. (I try to avoid 'correlated' / 'correlation' for the tail because it might not be linear).

Probably use this:

| c o r (X^{'}, Y^{'}) | ≫ 0

$|\cor(X', Y')| \gg 0$

where $X'$ is conditional on $X > 90\%$ of $X$ 's population, and $Y'$ is defined in the same sense.

correlation extreme-value Kmz
sumber

Independent variables that are dependent? My brain just exploded. You can't ask this sort of question on Monday morning

Aksakal

Given the upvoted answer, this Q does seem answerable.

gung - Reinstate Monica

To help this make sense to people, consider how much you care about gun issues and how much you like/hate the NRA. The correlation will probably be near zero. People who care the most about gun issues can either love or hate the NRA. But they will be very dependent. People who care the most about gun issues will almost never be in the middle of the pro-NRA/anti-NRA spectrum. People at the very top or bottom end of the pro-NRA/anti-NRA spectrum will tend to care more about gun issues than people in the middle.

David Schwartz

I'm sorry for stating the unclear question. I just want to visualize how it works for some independent distributions having some kinda of extreme dependence (not necessarily correlation).

Kmz

There are a host of copulas with weak overall dependence but strong tail dependence; the exact overall correlation would be affected by what the distribution of the marginals was.

Glen_b -Reinstate Monica

Jawaban:

Berikut ini contohnya $X$ dan $Y$ bahkan memiliki marginal normal.

Membiarkan:

X \sim N (0, 1)

$X \sim N(0,1)$

Bersyarat pada $X$ biarkan $Y = X$ jika $|X| > \phi$ , atau $Y = -X$ jika tidak, untuk beberapa konstanta $\phi$ .

Anda dapat menunjukkan itu, terlepas dari $\phi$ , sedikit kami memiliki:

Y \sim N (0, 1)

$Y \sim N(0,1)$

Ada nilai $\phi$ seperti yang $\text{cor}(X,Y) = 0$ . Jika $\phi = 1.54$ kemudian $\text{cor}(X,Y)\approx 0$ .

Namun, $X$ dan $Y$ tidak independen, dan nilai ekstrim keduanya sangat tergantung. Lihat simulasi di R di bawah, dan plot yang mengikuti.

Nsim <- 10000
set.seed(123)

x <- rnorm(Nsim)
y <- ifelse(abs(x)>1.54,x,-x)

print(cor(x,y)) # 0.00284 \approx 0

plot(x,y)

extreme.x <- which(abs(x)>qnorm(0.95))
extreme.y <- which(abs(y)>qnorm(0.95))
extreme.both <- intersect(extreme.x,extreme.y)

print(cor(x[extreme.both],y[extreme.both])) # Exactly 1

Chris Haug
sumber

(+1) If you want the distribution to not just be uncorrelated, but also not very dependent, you can do a modification of this that replaces the hard threshold swap with a fuzzy one. That's harder to get the math to line up, but it's doable.

Matthew Graves

Thank you Chris Haug! Your idea helps me visualize what I'm doing.

Kmz