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What is the meaning of the quantity:

$$\varepsilon=\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$$

Is there, for example, a geometric explanation? Is there a term for it in statistics?

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  • $\begingroup$ if the answer ok for you @apadana, can you please accept and/or upvote it? $\endgroup$
    – gunes
    Mar 26 '20 at 20:50
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It is the square root of the determinant of the covariance matrix (between $X$ and $P$). The determinant of the covariance matrix is called as Generalized Variance, which quantifies the co-variability of multivariate random variables to a scalar. What you write is the square root of it, so I believe it won't be too odd to call it as Generalized Deviation.

Edit: After some research, I found that, in some contexts, it's referred as Generalized Standard Deviation, or Wilk's standard deviation.

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  • $\begingroup$ Have you seen "generalized deviation" used anywhere, or did you come up with that term? $\endgroup$
    – Dave
    Mar 11 '20 at 18:22
  • $\begingroup$ @Dave I've come up with it actually. $\endgroup$
    – gunes
    Mar 11 '20 at 21:22
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    $\begingroup$ "Co-variability" is more accurate, than "variability". E.g. if the determinant of a covariance matrix is zero, there's linear dependence among the de-meaned random variables. (In the two by two case here, $\epsilon = 0$ if and only if $X$ lies in the linear span of $P$ and $1$.) $\endgroup$
    – Michael
    Mar 12 '20 at 1:28
  • $\begingroup$ Right @Michael changed it $\endgroup$
    – gunes
    Mar 12 '20 at 7:32

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