Horvitz Thompson Estimator :-S

Good afternoon!

I'm having some problems with a Horvitz Thompson estimator for the mean. The variance of this estimator to be exact.

My notes in class give the estimate of the variance of the estimator of the mean as the following:

V(that/N) = 1/N^{2 }{(SUM(h=1 to H) ( N_{h}^{2}/n_{h} ).(1 - (n_{h}/N_{h})).s_{h2 } }Where s_{h2} = (1/(n_{h} -1)) {SUM(i=1 to n_{h}) (y_{hi} - ybar_{h})^{2 }

This is fine. I understand how this works etc. The only problem is that my notes are the only place i've seen the equation written in this way. All the other material I've found on the way expresses everything to do with horvitz-thompson in terms of inclusion probabilities. For instance:

V(that/N) = 1/N^{2 }{(SUM(k in S)SUM(l in S) (1 - pi_{k}pi_{l}/pi_{kl} )(y_{k}y_{l }/ pi_{k}pi_{l})

I've tried to work out how one relates to the other, substituting n_{k}/N_{k} and n_{l}/N_{l} for the pi's in the second equation, but to no avail. My main problem is how you get to the s_{h2} in the top equation.

If anybody has any pointers, or has any idea where I can find an example of similar, it would be greatly appreciated.

Thankyou!