I have a sample of frequencies and Sums(totals):

Ni Si
23 12345
10 7800

Si is the sum over some variable X, measured Ni times

I build a variable Zi=Si/Ni (something as an X average by element)

If I compute the following moments over Z:

weighted variance, weighted kurtosis, or weighted asymmetry coefficients with proc univariate in SAS, with Ni being the weight variable,

I obtain good estimators of X variance, kurtosis and assym coefficients (I have made simulations to verify it).

The only problem is that I can not find references about that in internet.
I would like to know properties of these estimators (bias, variance, etc.).
I have already tried the keywords weighted moments, sample weighted moments, etc.
I also have looked to ratio estimators theory but got nothing.

I do not suppose finite population neither any information about distributions involved.

¿Any help for references?