Suppose the phase I sample is an unequal-probability sample of observations. If we observed yi for every unit in S (1), we could use the Horvitz– Thompson estimator of the variance of the Horvitz–Thompson estimator in (6.22) to estimate the first term in (12.14):

Where π (1)I = P(Zi = 1) and π (1) ik = P(ZiZk = 1) for i ≠ k and π (1) ii = P (Zi = 1).We need an estimator of V(ˆt (1) y ), however, that depends only on the y values in the phase II sample. Let π (2) ik = P (DiDk = 1 | Z) > 0. Show that

is an unbiased estimator of V(ṫ(1)y ).