For example, suppose I have a population partitioned into 8 strata (say, A - H), and I am interested in the proportion of the full population's total items of interest that are contained in 4 strata (say, A - D).

It is easy enough to get an estimate and variance for the denominator of the proportion (using conventional estimation procedures on all 8 strata); it is also easy enough to get an estimate and variance for the numerator of the proportion (using conventional estimation procedures on the 4 strata A - D).

My question is how to calculate the variance of the resulting proportion estimator. One might think about using Gaussian Error Propagation, but that requires the assumption that numerator and denominator of the proportion are independent when, in the case under consideration, the numerator estimate is derived from a subset of the strata that are the basis for the derivation of the denominator estimate.

Any thoughts would be appreciated. Thank you.