Sampling error with weighted mean

Question

I am studying statistics and I am wondering when it comes to standard error or a sampling if the calculation changes when there are weights added.

I have a weighted mean:

$$\mu_{w} = \dfrac{\sum_{i=1}^Nw_ix_i}{\sum_{i=1}^Nw_i}$$ and a weighted variance calculated by:

$$s^2_{w}=\dfrac{\sum_{i=1}^Nw_i}{(\sum_{i=1}^Nw_i)^2-\sum_{i=1}^Nw_i^2}\cdot \sum_{i=1}^N(x_i-\mu)^2$$

is the sampling error still calculated as

$$\text{SE}=\sqrt{\dfrac{s^2_{w}}{n}}$$

Answer 1

Unlike weighted mean and weighted variance which have definitions (although possibly multiple in the case of weighted variance), there does not seem to be a standardized definition for the weighted standard error of the mean (Gatz & Smith, 1995). Gatz and Smith give three examples in the paper. Some other sources would include this paper from Stata, perhaps.