Let a multivariate distribution be given by $P(Y,S_1,S_2)$, where all three variables are discrete, $Y$ is multivalued, $S_1=(0,1)$ and $S_2=(0,1)$, respectively, and all may be dependent. Define the quantity $X$ as
$$X=E(Y=y|S_1=1)-E(Y=y|S_2=1).$$
I estimate X by
$$\bar{x}=\frac{1}{\sum _{i=1}^{n}S_{1i}}\sum _{i=1}^{n} I(Y_i=y)S_{1i}-\frac{1}{\sum _{i=1}^{n}S_{2i}}\sum _{i=1}^{n} I(Y_i=y)S_{2i},$$
where $I$ is the indicator function and the sum is taken over a sample of size $n$, where $i=1,..,n$. Now find the variance of $\bar{x}$, $Var(\bar{x})$, in order to finally arrive at a s.e. estimate.
I would be thankful for any advice on how to approach this problem or a solution.