multi stage sampling without independence

I have a question regarding multi-stage sampling. To keep it simple, let's consider two stage sampling. Let p1 be the probability of picking a cluster at stage 1 using the Bernoulli sampling process. Now, if sub-sampling at the second stage is invariant and independent, that is, if the same sampling design is chosen for each cluster in the sample (invariance), and choosing elements within a cluster is independent of choosing elements in any other cluster (independence), then most statistics textbooks have the necessary formulas. However, if the sub-sampling processes were not independent, what would the formulas for an estimator and its variance be?

To give a more concrete example, let's consider the following 3 clusters which is the sample obtained from the first stage sampling process. Each element in the cluster is a pair <x, y>. I wish to estimate SUM(y) and there is no correlation between x and y values.

C1 = { <a, 10>, <b, 20>, <c, 10>}
C2 = {<b, 1>, <c, 5>, <d, 15>}
C3 = {<a, 4>, <c, 10>, <f, 23>, <g, 9>}

Let's say I have been given a random list from some other independent process: L = {a, c, g, h}.

Now, for sub-sampling at the second stage, I need to use the random list L (this list is created from another random process). It is used in the following way: For each element in each cluster , if x belongs to L, then is retained in the cluster. After sub-sampling, the resulting two stage sample is as follows:

C1 = { <a, 10>, <c, 10>}
C2 = { <c, 5>}
C3 = {<a, 4>, <c, 10>, <g, 9>}

As we can see, sub-sampling processes are not independent. The elements of all clusters are dependent through the random list L.

What estimator should I use for SUM(y) and how do I obtain the variance of the estimator? I could use the pi-estimator (also known as the Horvitz-Thompson estimator) at each stage, but I'm not sure about computing the variance computation.

Any pointers would be helpful.

Thank you very much!

• It's not clear to me how you are using L to select sub-units from within the PSU. Can you explain that more? – StatsStudent Jun 1 '17 at 6:39
• I have edited the question and explained how L is used to select sub-units. – Stat_Enthusiast Jun 1 '17 at 15:30

You should look at the Horvitz-Thompson estimator. For one thing, it is designed to compute sum(y). Its use isn't always easy. You need to compute the inclusion probability for each element (i.e. from C2), which is the probability $\pi_i$ that this element was included in the sample. You also should compute, for each pair of elements, a probability $pi_{ij}$ that element i and element j were included in the sample. If you can come up with these values (I don't think that should be too hard) you can get a H-T estimator of sum(y) along with its standard error.
The estimate of the total is $\hat\tau = \sum y_i/\pi_i$, and the estimate for the variance is widely available (but longer...), for instance,see equation 72 here, with equation 73 an alternative variance estimator.