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Let there be $i = 1,...,m$ random variables, for simplicity assume that each of these random variables follows a normal distribution: $x_{i} \sim N(\mu_{i}, \sigma_{i})$. Let $\hat{\rho_{i,j}}$ be the sample Pearson correlation coefficient between variable $x_{i}$ and $x_{j}$. Let X be a statistic of interest $$X = 1 + \sum_{i=1; i\neq j}^{n} \hat{\rho_{i,j}^{2}}$$

I need to find the expected value of $1/X$ and $1/X^{2}$, I would be fine with an approximation to them. I know that $\hat{\rho_{i,j}}$ has an exact distribution, but after looking at it it's honestly a little beyond my current skill set to find these expected values using the exact distribution. I have thought about the fact that Pearsons' correlation is approximately normal, and thus I have a sum of non-central chi-square distributions; however I'm concerned the sample size may not be sufficient for the normal approximation, and even considering it is valid I'm not sure how to proceed in this direction.

If it is simpler to use Kendall's tau (Spearman's etc) in place of Pearson's correlation coefficient that is acceptable as well, I just need $\hat{\rho_{i,j}}$ to be an estimate of the correlation between the two variables. Any guidance would be appreciated.

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    $\begingroup$ This question that I recently provided an answer to should be of interest: stats.stackexchange.com/questions/384897/… $\endgroup$ – StatsStudent Jan 2 at 21:13
  • $\begingroup$ Can you assume that $\rho_{ij}=0$. If you can, calculating the expectation becomes much simpler. $\endgroup$ – StatsStudent Jan 2 at 21:20
  • $\begingroup$ @StatsStudent unfortunately I can't assume that, the correlations will almost certainly not be 0. I'll take a look at that other questions, thanks! $\endgroup$ – Robert Montgomery Jan 2 at 21:28
  • $\begingroup$ All that needs to be done essentially is to find the PDF of $r$ and then use the Law of the Unconscious Statistician to find the expected value of any function of $r$. If I have some extra time tonight, I'll try to work it out and post. $\endgroup$ – StatsStudent Jan 2 at 21:29
  • $\begingroup$ If you can't assume $\rho=0$, I think this is going to be difficult to calculate exactly. Have you considered simulation? $\endgroup$ – StatsStudent Jan 2 at 22:13

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