Talk:Multivariate normal distribution

Infobox PDF requirement on covariance matrix

Previously the PDF said covariance matrix Σ had to be positive-definite. I fixed it to PSD for covariance matrix. I would also say symmetric but the article on PSD includes symmetric as part of the definition so I'll use that definition. Wqwt (talk) 20:53, 29 March 2022 (UTC)

Ok I misunderstood - it needs to be invertible thus positive definite to write down the PDF in that form, otherwise it is like zero variance Wqwt (talk) 21:22, 29 March 2022 (UTC)

More authoritative references?

User 174.168.4.26 has recently added three references to an SSRN preprint. The preprint is based on a student's Honors thesis (see link in section 4.3 of the preprint). There's presumably nothing wrong with the preprint but it might be preferable to cite more authoritative sources. Eldacan 08:15, 20 July 2023 (UTC)

Notation in Bivariate case

In the subsection titled "Bivariate case", the notation [XY]' is used twice. But that notation has not been previously defined and, in fact, is not used in the other sections of the article. Its meaning should be clarified. SometimesRPC 18:44, 8 January 2024 (UTC)

Formula for the limit of the isoline ellipsis' major axis in the bivariate case

The article's subsection "Bivariate case" states a formula for the limit of the major axis of isoline ellipses as $|\rho |\rightarrow 1$:

$y(x)=\operatorname {sgn}(\rho ){\frac {\sigma _{Y}}{\sigma _{X}}}(x-\mu _{X})+\mu _{Y}.$

and then provides a reference . However, the cited reference is just a formulation of the statement of the conditional expectation of the bivariate normal:

$\operatorname {E} (X_{1}\mid X_{2}=x_{2})=\mu _{1}+\rho {\frac {\sigma _{1}}{\sigma _{2}}}(x_{2}-\mu _{2})$

The reference discusses this within the context of estimation, which IMO complicates the statement unnecessarily. It seems more consistent to remove the reference and cite the Wikipedia article's own section.

M hoehle (talk) 09:19, 26 January 2024 (UTC)

Higher moments

There is a statement 'the expected value μ is taken to be 0 in the interests of parsimony' in the section Properties. Higher moments. On the Wikipedia site Isserlis' theorem there is a statement that this theorem holds for a zero-mean multivariate normal random vector. It implies that either the expected value μ is taken to be 0 not in the interests of parsimony, but it is the substantial assumption of the Isserlis' theroem and thus one cannot sum the products of covariances in general so the text needs to be corrected or other source is needed for the generalisation of the Isserlis' theorem for a non-zero-mean multivariate normal random vector which allows to sum the products of covariances (if that generalisation exists). 91.94.10.76 (talk) 16:32, 11 March 2025 (UTC)

Right, that's confusing as written. I'd simply delete the part that says "(the expected value μ is taken to be 0 in the interests of parsimony)". What it is failing to properly say is the the formula for central moments is the same as the formula for non-central moments in the case that one simply deletes from the latter all terms that depend upon a simple mean. —Quantling (talk | contribs) 16:52, 11 March 2025 (UTC)