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 Ï€A I = P (i ˆˆ SA) and sampling weights wAi = 1/Ï€Ai . Corresponding quantities for frame B are SB, Ï€B i , and wB i . Let δi = 1 if unit i is in domain ab and 0 otherwise. Then ˆtAa = Æ©iˆˆSA wBi (1 ˆ’ δi)yi and ṫBb = Æ©iˆˆSBwBi (1 ˆ’ 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 =Æ©iˆˆSAwAi δiyi andˆtBab =Æ©iˆˆSBwBi δiyi.
a. Let θ ˆˆ [0, 1]. Show that

is an unbiased estimator of ty = Æ©Ni=1 yi with

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

y0 = + e, + (1 - e, + !! +(1 e) ab ab