Sampford's unequal-probability sampling without replacement

With target inclusion probabilities \(\pi_i\) (\(0 < \pi_i < 1\), summing to an integer n) the rejective procedure draws one unit with probability \(\pi_i / n\) and n - 1 further units with replacement with probabilities proportional to \(\pi_i / (1 - \pi_i)\), accepting the first attempt whose n units are distinct; the inclusion probabilities are then exactly \(\pi_i\) (Sampford 1967). The joint inclusion probabilities follow Sampford's formula through the elementary symmetric functions of \(\lambda_i = p_i / (1 - n p_i)\), \(p_i = \pi_i / n\). Units with \(\pi_i\) within 1e-6 of 0 or 1 are fixed out or in. Every attempt uses n Philox uniforms from stream attempt, so the R and Python arms draw the same sample.

Usage

SampfordDesign(pik, seed = 0, max_iter = 500L)

Arguments

pik

Inclusion probabilities summing to an integer n >= 2.

seed

Philox key.

max_iter

Maximum rejective attempts.

Value

List with sample (0/1 indicator), joint (matrix of joint inclusion probabilities, \(\pi_i\) on the diagonal), attempts and n.

References

Sampford, M. R. (1967). On sampling without replacement with unequal probabilities of selection. Biometrika 54, 499-513.

Tille, Y. (2006). Sampling Algorithms. Springer, Sec. 7.6.

Examples

SampfordDesign(c(0.2, 0.4, 0.6, 0.8))$sample
#> [1] 0 0 1 1