Brewer€™s (1963, 1975) procedure for without-replacement unequal probability sampling.
For a sample of size n = 2, let πi be the desired probability of inclusion for
psu i, with the usual constraint that

And

Draw the first psu with probability ai /Æ©Nk =1 ak of selecting psu i. Supposing psu i is selected at the first draw, select the second psu from the remaining N ˆ’ 1 psus with probabilities ψj / (1 ˆ’ ψi).
a. Show that

b. Show that P (psu i selected in sample) = πi.

c. The SYG estimator of V (ṫHT) for one-stage sampling is given in (6.23). Show that Ï€iÏ€j ˆ’ Ï€ij ‰¥ 0 for Brewer€™s method, so that the SYG estimator of the variance is always nonnegative.

ai = ak ! 4-1 !