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 πA I = P (i ∈ SA) and sampling weights wAi = 1/πAi . Corresponding quantities for frame B are SB, πB i , and wB i . Let δi = 1 if unit i is in domain ab and 0 otherwise. Then ˆtAa = Ʃi∈SA wBi (1 − δi)yi and ṫBb = Ʃi∈SBwBi (1 − δi)yi estimate the domain totals ta and tb, respectively. There are two independent estimators of the population total in the intersection domain ab:ˆtAab =Ʃi∈SAwAi δiyi andˆtBab =Ʃi∈SBwBi δiyi.
a. Let θ ∈ [0, 1]. Show that

is an unbiased estimator of ty = ƩNi=1 yi with

b. Show that V (ṫy,θ) is minimized when