I measure a series of $n$ objects [O_1, O_2, O_3, ..., O_n]. Because those measurements are quite hard to perform, I have quite a lot of measurement error and therefore I measured each object $r$ times to get a better estimate. The distributions of measurements are bounded between 0 and 1 (inclusive) but are not uniform. Might kinda look like a normal (but bounded) distribution, eventually skewed though. The standard deviation of the measurements for the objects O_i is $sd_i$.
I want to compute the geometric mean of the mean measurement of those objects
$$X = \left(\prod \bar O_i\right)^{\frac{1}{n}}$$
, where $\bar O_i$ is the mean of the $r$ measurements of the object $O_i$.
What is the standard deviation and the standard error of the geometric mean of these $n$ variables?
Is the standard deviation of $X$ simply
$$sd_X = \left(\prod sd_i\right)^{\frac{1}{n}}$$
and the standard error of $X$ ($se_X$) would simply be
$$se_X = \frac{\left(\prod sd_i\right)^{\frac{1}{n}}}{r}$$