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 ν_h = m_h / n_h, h = 1, . . . ,H be the proportion of phase I units in stratum h to be sampled in phase II.

1. Show that the expected cost is (12.16).

2. Show that V(y̅  ⁽²⁾_str ) is minimized, subject to the constraint in (12.16), when ν_h is given in (12.17).