Variance-minimising shares for a choice-based sample
Source:R/horowitz_native4.R
morie_choice_based_shares.Rd\(q_0 = \sqrt{\pi_0}/(\sqrt{\pi_0} + \sqrt{\pi_1})\) and
\(q_1 = \sqrt{\pi_1}/(\sqrt{\pi_0} + \sqrt{\pi_1})\). The
designer of a choice-based sample picks the stratum sizes, and the
asymptotic distribution depends on them only through
\(\pi_1/q_1 + \pi_0/q_0\). Minimising that factor gives the
square-root rule: the optimum is NOT \(q_j = \pi_j\) (a random
sample) and not an even split either, except at \(\pi_1 = 1/2\)
where the two coincide. Mirrors morie.fn.hrzcbsm.