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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:

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The only problem with the formula is I'm not very sure how to calculate the joint sampling probability.

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This page gives me:

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but this page gives me:

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What to do? There is an answer that the probability is context specific, but why?

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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.

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  • $\begingroup$ Thanks for the info. I'm interested in estimating a population total in a weighted complex survey with stratification. I'm not sure why the joint probability is context specific. Furthermore, how does a survey analysis software such as SPSS deal with the variance? I can't find useful info but I'll look at the paper. $\endgroup$ – SmallChess Jun 28 '15 at 1:53
  • $\begingroup$ The thing is that the HT estimator can also be used e.g. when using multistage sampling, a combination of multistage sampling and stratification, sampling with or without replacement, ... So in each of these designs the "formulas" of the joint probabilities will be different and depend on different info (e.g. the joint inclusion probability in SRS is different when using sampling with vs without replacement). Anyhow, I have no experience with SPSS so I can't help you with that. $\endgroup$ – Jorne Biccler Jun 28 '15 at 9:26
  • $\begingroup$ Thanks for the hint. Is there like a paper, book or anything that you can recommend me? This is a quite advanced topic that I've finding hard to find anything on the Internet. $\endgroup$ – SmallChess Jun 28 '15 at 12:05

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