The estimators of V (HT) in (6.22) and (6.23) require knowledge of the joint inclusion probabilities ik.

The estimators of V (ṫHT) in (6.22) and (6.23) require knowledge of the joint inclusion probabilities πik. To use these formulas, the data file must contain an n × n matrix of the πik’s, which can dramatically increase the size of the data file; in addition, computing the variance estimator is complicated. If the joint inclusion probabilities πik could be approximated as a function of the πi’s, estimation would be simplified. Let ci = πi(1 − πi). Hájek (1964) (see Berger, 2004, for extensions) suggested approximating πik by

a. Does the set of ˜πik’s satisfy condition (6.18)? Can they be joint inclusion probabilities?
b. What is ˜πik if an SRS is taken? Show that if N is large, ˜πik is close to πik .
c. Show that if ˜πik is substituted for πik in (6.21), the expression for the variance can be written as

Where ei = ti /πi − A and

Write (6.21) as

d. We can estimate ṼHaj(ˆtHT) by

Where

That if an SRS of size n is taken, then ṼHaj(ṫHT) = N2(1 − n/N)s2 t .