Talk:Elliptical distribution

Is this statement always true?
I am not sure the following statement is always true:

"Elliptical distributions are important in portfolio theory because if the returns on all assets available for portfolio formation are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance..."


 * I think a multivariate Cauchy distribution is elliptical, but mean and variance would necessarily be undefined. Rlendog (talk) 19:53, 14 December 2010 (UTC)

This statement is wrong since Levy-Stable distributions are elliptical and only the Gaussian (special case) has a variance. Limit-theorem (talk) 00:44, 8 August 2014 (UTC)


 * (davidma) I agree with the above. You can have many different elliptical distributions with the same mean and variance. — Preceding unsigned comment added by 67.161.7.231 (talk) 04:57, 15 July 2011 (UTC)


 * The quote from the article missed out the bit "two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return", which may have some effect (but the meaning isn't defined). Still, it looks as if there should be a change from "variance" to "scale paramerter" at least, and presumably there should also be something like "... for a given characteristic generator...". Has anyone looked at the two references for this paragraph to see what is going on there? Melcombe (talk) 09:07, 15 July 2011 (UTC)


 * This article needs some work. Most article define the spherical distribution first, with the elliptical as a special case, and the characteristic generator has to have properties clearly discussed here. I will work on it soon. Limit-theorem (talk) 00:43, 8 August 2014 (UTC)

Constraints on g
I don't have the paper cited available right now, but it seems strange to me that in the definition g should be constrained to the domain of non-negative reals. In the case of a normal distribution, the domain for exp is actually constrained to the nonpositive reals. Can anyone verify that this is correct/incorrect?


 * It is correct in the article. $$(x-\mu)\Sigma^{-1}(x-\mu)$$ will always be nonnegative real since $$\Sigma$$ is p.d. For a normal distribution we have a minus included as part of the g function, so it's $$g(\eta)=\exp(-\frac{1}{2}\eta)$$.  Some authors like the Landsman Valdez (2003) linked in the article also write it in a slightly different way, with a $$\frac{1}{2}$$ included and $$k$$ also written slightly differently.    — Preceding unsigned comment added by 160.39.81.41 (talk) 04:18, 31 August 2016 (UTC)

Removed portfolio section
I removed this, because of the problems already discussed and others.


 * Economics and business

Elliptical distributions are used in portfolio theory. If the returns on all the assets available for portfolio formation are jointly elliptically distributed, then all portfolios can be characterized completely by their location and scale – that is, any two portfolios with identical location and scale of portfolio return have identical distributions of portfolio return (Chamberlain 1983; Owen and Rabinovitch 1983). For multivariate normal distributions, location and scale correspond to mean and standard deviation. 162.250.169.162 (talk) 17:11, 28 August 2016 (UTC)

Evaluation
I added the math project template and upped the stats box importance to mid. 162.250.169.162 (talk) 13:36, 31 August 2016 (UTC)

Variance of sperical distribution
In general the variance of a spherical distribution does not have to be the identity; it can be in the form constant * identity matrix. I edited the section to fix this — Preceding unsigned comment added by 84.226.196.209 (talk) 12:22, 8 May 2017 (UTC)