It's a somewhat complicated situation and sorry about my phrasing, but it's my first time here. Suppose I have random normal variable $X$ ~ $N( \mu, \sigma^2)$, which represents some true effect(s). Then I try to estimate the mean of this effects using the estimates from a few (sometimes underpowered) trials -> Meta-Analysis, and do the following transformation:
$Y$ = $c/X^2$, where $c$ is a constant $> 0$ and $Y$ must be $> 0$ and therefore x>0. As a side note, $\sigma$ is for example 90% of $\mu$ and divided by the 97.5%-quantile of the standard normal distribution. What distribution has $Y$? But most importantly how can I determine/estimate the central tendency of a sample from this resulting distribution?
What I found out til now, just squaring that $X$ would result in a non-central chi-square distribution with a non centrality parameter and then just do the inverse (maybe?) and still not knowing the central tendency. Also with that 90%-$\sigma$, I have some x<0 and with very small x I get a super long tail (which doesn't reflect any reality anymore, it's just a theoretical number). Tested with $\mu=6$ and $\sigma=2.76$, c=11128.6 and 3 trials 99%,80% and 10% powered depending on the trial size in a simulation with n=10000.
I worked with the Median, but it doesn't seem to catch the center well.