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I am interested in the moments, we have for instance the mean, $\mathrm{E}(X)$ and $\mathrm{E}(X^2)$. What about values like $\mathrm{E}(X^{1.5})$ or $\mathrm{E}(X^{-1})$? Have they been investigated?

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    $\begingroup$ Stuart & Ord, in Kendall's Advanced Theory of Statistics (Fifth Ed.), section 3.26, discuss such moments and provide references. They show how to obtain negative and fractional moments of positive distributions from their moment-generating functions (provided those are defined for all non-positive arguments). One reference discusses relationships between fractional moments and fractional derivatives of the characteristic function. $\endgroup$
    – whuber
    Commented Oct 28, 2012 at 15:32

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Yes, investigated to at least some extent, as is readily seen by googling 'inverse moment' or 'fractional moments'.

Edit: In some cases these moments are rather straightforward to calculate. Here's an example of computing $E(X^{3/2})$ for $X\sim\text{gamma}(\alpha,1)$:

\begin{eqnarray} E(X^{3/2}) &=& \int_0^\infty x^{3/2} f(x) dx \\ &=& \frac{1}{\Gamma(\alpha)} \int_0^\infty x^{3/2} x^{\alpha-1} e^{-x} dx\\ &=& \frac{\Gamma(\alpha+3/2)}{\Gamma(\alpha)}\cdot \frac{1}{\Gamma(\alpha+3/2)} \int_0^\infty x^{(\alpha+3/2)-1} e^{-x} dx\\ &=& \Gamma(\alpha+3/2)/\Gamma(\alpha) \end{eqnarray}

You can as easily do $E(X^{-1})$ (as long as $\alpha>1$).

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