There is a Wikipedia page for the Horvitz–Thompson estimator. It is an estimator for the population total. Unfortunately the page has failed to state the standard error.
From here, the standard error is:
The only problem with the formula is I'm not very sure how to calculate the joint sampling probability.
This page gives me:
but this page gives me:
What to do? There is an answer that the probability is context specific, but why?
Best Answer
First of all, the variance formula you found is indeed correct. The calculations of the 'second-order' inclusion probabilities depend largely on the specific sampling design you are using. Since the Horvitz-Thompson estimator (and the corresponding variance estimator) is an estimator that can be applied in a large range of sampling designs a general formula for these probabilities cannot be expected. Thus depending on the design you're using one of the formulas could be 'correct' or both could be 'wrong'.
If you can't find additional information on the internet you should perhaps try to calculate them yourself? Some examples can be found in the original paper "A Generalization of Sampling Without Replacement From a Finite Universe" by Horvitz and Thompson. If you would still need help perhaps you should provide some additional info.