# Horvitz-Thompson variance estimation when estimating across strata

I have a sample of Business units, which has been stratified according to two stratification variables (Revenue class and field of Business acitivity). Within the strata, Units were sampled according to simple random sampling without replacement and after a specified sample size for each stratum, hence inclusion probabilities of Units of the same stratum are identical.

The business Units are from different geographical regions which have not been considered when building strata. Now I want to build estimates for each Region. Hence, within the estimation strata (regions) are Units with unequal probabilities.

I assume that the Horvitz-Thompson estimator might be appropriate for this purpose. However, when it Comes to the estimation of the variance of this estimator,

I am not sure how to calculate the Joint inclusion probability. Does anyone know how to do it according to the explained stratification scheme? Or would there even be another estimator that would be more appropriate? Let me know if you kneed further Information. Thank you a lot in advance!

Horvitz Thompason is a general variance formula for non-replacement unequal sampling. The joint probabilities are usually unknown and must be estimated. That means Horvitz Thompason is not used directly (unless you really know the joint probabilities).

Some possible methods:

1. Hartley–Rao estimates the joint-probability as

2. Linearization estimation

3. Bootstrap

4. Replication methods

https://cran.r-project.org/web/packages/survey/vignettes/pps.pdf (only the first and second page) has some details.