(Solution) - Consider the dual frame survey in Figure 14 1 b in which -(2025 Original AI-Free Solution)

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Consider the dual frame survey in Figure 14.1(b) in which independent probability samples are taken from frames A and B. Suppose that all three domains are nonempty. Let SA denote the sample from frame A, with inclusion probabilities ?A I = P (i ? SA) and sampling weights wAi = 1/?Ai . Corresponding quantities for frame B are SB, ?B i , and wB i . Let ?i = 1 if unit i is in domain ab and 0 otherwise. Then ?tAa = ?i?SA wBi (1 ? ?i)yi and ?Bb = ?i?SBwBi (1 ? ?i)yi estimate the domain totals ta and tb, respectively. There are two independent estimators of the population total in the intersection domain ab:?tAab =?i?SAwAi ?iyi and?tBab =?i?SBwBi ?iyi.
a. Let ? ? [0, 1]. Show that

is an unbiased estimator of ty = ?Ni=1 yi with

b. Show that V (?y,?) is minimized when