Optimal allocation for two-phase sampling with stratification.
Suppose phase I is an SRS and phase II is a stratified random sample, and that the total cost for the sample is given in (12.15), where c(1) is the cost to sample a unit in phase I and ch is the cost to sample a unit in stratum h in phase II. Let νh = mh / nh, h = 1, . . . ,H be the proportion of phase I units in stratum h to be sampled in phase II.
a. Show that the expected cost is (12.16).
b. Show that V( (2)str ) is minimized, subject to the constraint in (12.16), when νh is given in (12.17).