Kalton and Anderson (1986) consider disproportional stratified random sampling for estimating the mean of a characteristic yi in a rare population. Let ri = 1 if person i is in the rare population and 0 otherwise. Stratum 1 contains N1 persons, M1 of whom are in the rare population; stratum 2 contains N2 persons, with M2 persons in the rare population. We wish to estimate the population mean Ud = ƩNi =1 riyi / (M1+M2) using a stratified random sample of n1 persons in stratum 1 and n2 persons in stratum 2.
a. Suppose A = M1 / (M1 + M2) is known. Let d = A1 + (1 − A) 2, where 1 and 2 are the sample means of the rare population members in strata 1 and 2, respectively. Show that, if you ignore the finite population corrections (fpcs) and if the sampled number of persons in the rare population in each stratum is sufficiently large, then

Where S2 j is the the variance of y for the rare population members in stratum j and
pj = Mj / Nj for j = 1, 2.
b. Suppose that S21 = S22 and that the cost to sample each member of the population is the same. Let f2 = n2/N2 be the sampling fraction in stratum 2, and write the sampling fraction in stratum 1 as f1 = kf2. Show that the variance in (a) is minimized for a fixed sample size n when k = √ p1/p2.