E1 Design effect and effective sample size

To get the design effect formula in Theorem 3.1, notice that the sampling variance is

while the variance of simple random sampling is p(1 - p)/n. Thus,

The formula for the total effective sample size is then easily obtained:

E2 Weighting effect

To prove Theorem 3.2, we consider x1, …, xn as independent identically distributed random variables. Because the weights are constant we obtain

Therefore, The design effect is, by definition, the actual variance divided by the variance of simple random sampling which is var(x)/n. Consequently,

The first property of the calibrated sample size has been proved in [10] but we give another, much shorter, proof. By the Cauchy—Schwarz inequality, for any real numbers (ai), (bi) with the equality if and only if there is ë such that ai = λbi for all i. (This inequality, known also as the Cauchy—Bunyakovskij inequality, is discussed in almost any textbook on functional analysis or Hilbert spaces.) Therefore,

so that with the equality if and only if all weights are constant.

To prove the second property, assume that n = n1 + … + nm. We have to show that

It is enough to prove this inequality in the case of two summands because then the repetition gives the general case.