Double Sampling
Double Sampling was written by Michael Hidiroglou in 2001. It was published in Survey Methodology (vol. 27, no. 2).
The theory of double sampling, or two-phase sampling, is introduced. The core concept is that there are two samples, one nested in the other. Notation used is:
population U = {1, ..., N}
first sample s1(s1 ⊆ U)
first sample inclusion probability for unit k: π1k = P(k ∈ s1)
first sample weight w1 = 1/π1k
second sample s2(s2 ⊆ s1 ⊆ U)
second sample inclusion probability for unit k, given the first sample:
second sample weight w2 = 1/π2k
overall sample weight w*k = w1 w2
The author then derives estimators that can be used in both nested and non-nested cases.
Unfortunately this math is a bit beyond me right now.