mean of an absolute Student’s t

Posted on November 30, 2011 by xi'an in R bloggers | 0 Comments

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Having (rather foolishly) involved myself into providing an answer for Cross Validated: “Can the standard deviation of non-negative data exceed the mean?“, I ended up having to derive the mean of the absolute value of a Student’s variate X. (Well, not really, but then I did.) I think the following is correct:

$\mathbb{E}[|X|]=\mu(2\text{P}_{\mu,\nu}(X>0)-1)\qquad\qquad\\ \qquad\qquad 2 f(0,\nu)\dfrac{\nu}{\nu-1}\left[1 \mu^2/\nu\right]^{-(\nu-1)/2}$ 0)-1)\qquad\qquad\\ \qquad\qquad+2 f(0,\nu)\dfrac{\nu}{\nu-1}\left[1+\mu^2/\nu\right]^{-(\nu-1)/2}’ title=’\mathbb{E}[|X|]=\mu(2\text{P}_{\mu,\nu}(X>0)-1)\qquad\qquad\\ \qquad\qquad+2 f(0,\nu)\dfrac{\nu}{\nu-1}\left[1+\mu^2/\nu\right]^{-(\nu-1)/2}’ class=’latex’ />

where $f(x,\nu)$ is the density of the standard Student’s distribution. (I also checked by simulation. The derivation is there. And now fully arXived. Even though it is most likely already written somewhere else. See, for instance, Psarakis and Panaretos who studied the case of the absolute centred t rv, called the folded t rv.)