Assume we have two $p\times 1$ random, zero-mean vectors $X$ and $Y$ with covariance matrices: $E[XX^T]=Q$ and $E[YY^T]=S$, and also $E[YX^T] = R$.

Also, let $B$ be an $n\times p$ matrix.

I want to find the matrix $A$ that maximizes the expected value of the Kendall rank correlation between the vectors $AX$ and $BY$.

In other words, I want to find the matrix $A$ such that the ordering of the elements of $AX$ and $BY$ is as similar as possible.

My guess is that the solution should be something like $A=BRQ^{-1}$ (i.e., projecting $X$ first to $Y$ space, and then use $B$). However, I cannot prove that this guess is actually maximizing the expected Kendal correlation.


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