Joint Inclusion Probabilities using Brewer's Approximation

Computes approximate joint inclusion probabilities using equation (18) from Brewer & Donadio (2003).

up_brewer_jip(pik)

Arguments

pik

numeric vector of first-order inclusion probabilities.

Value

A symmetric N×N matrix of approximate joint probabilities.

Details

Uses the approximation: $$c_i = \frac{n-1}{n - \frac{2n-1}{n-1}\pi_i + \frac{\sum \pi_k^2}{n-1}}$$ $$\tilde{\pi}_{ij} = \pi_i \pi_j \frac{c_i + c_j}{2}$$

This has the best model-assisted properties among Brewer's approximations.

References

Brewer, K.R.W. and Donadio, M.E. (2003). The High Entropy Variance of the Horvitz-Thompson Estimator. Survey Methodology, 29(2), 189-196.

Examples

pik <- c(0.2, 0.3, 0.5)
pikl <- up_brewer_jip(pik)