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I have a question about the definition of the median of a (continuous) random vector $X\equiv (X_1,..., X_r)$. As suggested here and by other discussions in this forum (e.g., here and here) there are various definitions of median for the multivariate case.

Suppose that $$ (A) \quad \Pr(X_1\geq 0,..., X_r\geq 0)=\Pr(X_1\leq 0,..., X_r\leq 0)=0.2 $$

Is there any uncontroversial relation between (A) and the notion of "median equal to zero", independently of the definition of multivariate median one could use?

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    $\begingroup$ There is no reason to believe (A) implies the "marginal median" is $0$ and similarly with the other multivariate medians. $\endgroup$
    – Henry
    Nov 10, 2021 at 10:08

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