I've found a related question which was for product moment $E[XY]$. Is $E[XX^T]$ also called product moment or is there a special case name for it? This is the first term of the covariance matrix $Cov(X) = E[XX^T] - E[X]E[X]^T$.
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$XX^T$ is called an outer product, if that helps. – StubbornAtom Dec 05 '21 at 09:40
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Thanks, I guess some it's some "outer product moment" – stevew Dec 05 '21 at 10:21
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I think the notation should be the opposite: $X$ typically has variables in columns and observations in rows, the opposite of what you have used. – Richard Hardy Dec 05 '21 at 13:00
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I don't know if there exists a special name for $E(XX^\top)$ but since \begin{align} XX^\top=\begin{pmatrix} X_1 \\ \vdots \\X_n\end{pmatrix}\cdot \begin{pmatrix} X_1 & \dots & X_n\end{pmatrix}=\begin{pmatrix} X_1^2 & \dots & X_1X_n \\ \vdots & \ddots & \vdots \\ X_nX_1 & \dots & X_n^2 \end{pmatrix} \end{align} you get: $$ E(XX^\top)= \begin{pmatrix} E(X_1^2) & \dots & E(X_1X_n) \\ \vdots & \ddots & \vdots \\ E(X_nX_1) & \dots & E(X_n^2) \end{pmatrix} $$ Therefore, I think that it is valid to call this matrix the matrix of product moments, or product moment in a multivariate setting.
Lars
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I think the notation should be the opposite: $X$ typically has variables in columns and observations in rows, the opposite of what you have used. – Richard Hardy Dec 05 '21 at 13:00
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1Well, I thought about $X=(X_1,\dots,X_n)^\top$ being a column vector. Isn't this the usual notation ? – Lars Dec 05 '21 at 13:41
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