weighted empirical cdf consistency/unbiasedness?

Question

The weighted empirical distribution function estimator is given by:

$\hat{F}(x)=\frac{1}{\sum_{i=1}^{n}w_{i}}\sum_{i=1}^{n}w_{i}I(X_{i}\leq x)$

see also here: http://www.okstate.edu/sas/v8/sashtml/insight/chap38/sect25.htm

I fail to find proofs/discussions of the properties of this estimator. Is this estimator an unbiased estimator of $F(x)$? Or at least asymptotically unbiased? Or maybe consistent? Why? What assumptions are needed in order for it to be? You can assume anything you want, I am looking for a discussion.

Thanks.

Answer 1

If $w_i \geq 0$ with atleast one $w_i >0$, $X_i \sim F$ are identically distributed but not necessarily independent then your estimator is unbiased

$$ \begin{align} \mathbb{E}[\hat{F}(x)] &=\sum_{i=1}^{n}\frac{w_i}{\sum_{i=1}^n w_i}\mathbb{E}[I(X_{i}\leq x)] \\ &= \sum_{i=1}^{n}\frac{w_i}{\sum_{i=1}^n w_i}F(x) \\ &= F(x) \end{align} $$

The first line comes form linearity of expectation operator, second as the random variables are identically distributed and last because the sum of re-weighted weights is 1.