$X$ is a random variable with unknown distribution. A number of experiments are conducted to estimate $X$. Each experiment has a different reliability measure in estimating $X$. These $n$ experiments resulted in following sample set $\{x_1, x_2, x_3, ... , x_n\}$ with corresponding non-zero weights being $\{w_1, w_2, w_3, ... , w_n\}$. The higher weight corresponds to higher reliability. Note ${\sum_{i=1}^n{w_i}}$ can be greater than $1$.
The best unbiased estimator of true value of $X$ is the weighted mean of sample,
$\hat{X} = \bar{x}_w$,
where, $\bar{x}_w = \frac{\sum_{i=1}^n{w_ix_i}}{\sum_{i=1}^n{w_i}}$
The estimator for variance of $X$ from its true mean is,
$\hat{\sigma^2} = \bar{\sigma^2}_w$,
where, $\bar{\sigma^2}_w = \frac{\sum_{i=1}^n {w_i(x_i-\bar{x}_w)^2}}{\sum_{i=1}^n{w_i}}$,
What would be the best estimate of Standard Error of the sampling distribution of $\bar{x}_w$. Would it be $\frac{\bar{\sigma}_w}{\sqrt{n}}$. If yes, can someone help derive/explain it.
