In simple random sampling, we know that a without-replacement sample of size n has smaller variance than

Question:

In simple random sampling, we know that a without-replacement sample of size n has smaller variance than a with replacement sample of size n. The same result is not always true for unequal-probability sampling designs (Raj, 1968, p. 56). Consider a with replacement design with selection probabilities Ïˆi, and a corresponding without replacement design with inclusion probabilities Ï€i =nÏˆi; assume nÏˆi i = 1. . . N.
a. Consider a population with N = 4 and t1 = ˆ’5, t2 = 6, t3 = 0, and t4 = ˆ’1. The joint inclusion probabilities for a without-replacement sample of size 2 are Ï€12 = 0.004, Ï€13 = Ï€23 = Ï€24 = 0.123, Ï€14 = 0.373, and Ï€34 = 0.254. Find the value of Ï€i for each unit. Show that for this design and population, V (á¹«Ïˆ) b. Show that for Ï€i = nÏˆi and V (á¹«Ïˆ) in (6.8),

c. Using V (á¹«HT) in (6.21), show that if

Then V (á¹«HT) ‰¤ V(á¹«Ïˆ).
d. Gabler (1984) shows that if

Then V (á¹«HT) ‰¤ V (á¹«Ïˆ). Show that if Ï€ik ‰¥ (n ˆ’ 1) Ï€iÏ€k / n for all i and k, then
Gabler€™s condition is met.
e. (Requires knowledge of linear algebra.) Show that if V (á¹«HT) ‰¤ V (á¹«Ïˆ), then