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Suppose that there are two sample covariances $s_{jk}$ and $s_{lm}$ obtained from $n$ observations following normal distribution that satisfy $E(s_{jk})=\sigma_{jk}$ and $E(s_{lm})=\sigma_{lm}$.

Then, what is the exact expression for $E(1/s_{jk}^2)$ or $E(1/(s_{jk}s_{lm}))$ or $E(1/(s_{jk}^2s_{lm}^2))$?

Although the result for $E(1/s_{jk})$ is available in some books/papers, I cannot for the above cases. Are there any reference that provides the results?

Thank you.

kjetil b halvorsen
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user0131
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    I think you should define: (a) the estimator of sample covariance you are using, (b) change "normal distribution" to jointly bivariate Normal, and specify the variance covariance matrix you are using (basic notation), (c) provide links to the papers/books you claim provide an exact solution to the expectation of the inverse. – wolfies Oct 02 '17 at 06:24
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    Since (assuming a nondegenerate joint Normal distribution and $j\ne k$) those sample covariances have nonzero density at zero, the moments of any expression involving their reciprocals will be undefined or infinite. See https://stats.stackexchange.com/questions/299722. – whuber Oct 02 '17 at 15:10
  • ... which is also why I am curious to see the claimed results in books/papers ;) – wolfies Oct 02 '17 at 17:30

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