Suppose you want to take a sample of students in a university but your sampling frame is a list of all classes offered by the university. A student may be in more than one class, so a probability sample of classes, which includes all students in those classes, may contain some students multiple times. Lavallée (2007) describes a generalized weight share method for such situations, and this exercise is adapted from results in his book. Let UA be the sampling frame population with N units. Let Zi = 1 if unit i is in the sample SA and 0 otherwise, with πi = P (Zi = 1). The target population UB has M elements. Each element in UB is linked with one or more of the units in UA; let

and let

We assume Lk ≥ 1 for each k and that Lk is known. In our
example, ℓik = 1 if student k is in class i and Lk is the number of classes taken by student k. Let yk be a characteristic associated with element k of UB. We want to
estimate

a. Let

Show that ṫy is an unbiased estimator of ty, with

b. Let SB be the set of distinct units sampled from UB using this procedure. Show that ṫy can be rewritten as

We can view

as a “weight” for yk .
c. If Lk = 1 for all k, show that ṫHT in (6.19) is a special case of ṫy. What is w∗ k in this case?
d. Suppose UA = {1, 2, 3}, UB = {1, 2} and the values of ik are given in the following table:

Suppose y1 = 4 and y2 = 6, so that ty = 10. Find the value of ṫy for each of the three possible SRSs of size 2 from UA. Using the sampling distribution of ṫy, show that ˆty is unbiased but that V (ṫy) > 0. Even though each possible SRS from UA contains both units from UB (so in effect, a census is taken of UB), the variance of ṫy is not zero.
e. Data file wtshare.dat contains information from a hypothetical SRS of size n = 100 from a population of N = 40,000 adults. Each adult in the sample is asked about his or her children: how many children between ages 0 and 5, whether those children attend preschool, and how many other adults in the population claim the child as part of their household. Estimate the total number of children in the population who attend preschool. Use the with-replacement variance estimator

to construct an approximate 95% CI for the total number of children who attend preschool.