Pairwise Expectations for Chromy Sampling

Estimates pairwise hit expectations \(E(n_i n_j)\) for Chromy's sequential PPS sampling via Monte Carlo simulation.

up_chromy_pairexp(x, n, nsim = 10000L)

Arguments

x

Numeric vector of positive size measures.

n

Sample size.

nsim

Number of simulations (default 10000).

Value

A symmetric N x N matrix of pairwise expectations. Entry (i, j) is \(E(n_i n_j)\), the expected product of hit counts. Diagonal entries are \(E(n_k^2)\).

Details

Chromy's method is a minimum replacement (PMR) design where units with large size measures can be selected multiple times. The appropriate variance estimator uses pairwise expectations rather than joint inclusion probabilities.

Chromy (2009) gives the generalized Yates-Grundy variance: $$V(\hat{T}) = \frac{1}{2} \sum_{i \neq j} \{E(n_i)E(n_j) - E(n_i n_j)\} \left(\frac{y_i}{E(n_i)} - \frac{y_j}{E(n_j)}\right)^2$$

where \(E(n_k) = n x_k / \sum x\) is exact.

In the without-replacement case (all \(E(n_k) < 1\)), this reduces to the standard Sen-Yates-Grundy formula with \(E(n_i n_j) = \pi_{ij}\).

References

Chromy, J.R. (2009). Some Generalizations of the Horvitz-Thompson Estimator. Proceedings of the Survey Research Methods Section, ASA, 216-227.

Chauvet, G. (2019). Properties of Chromy's sampling procedure. arXiv:1912.10896.

See also

up_chromy() for sampling, up_brewer_jip() for joint inclusion probabilities under high-entropy WOR designs.

Examples

x <- c(10, 20, 15, 25, 30)
pairexp <- up_chromy_pairexp(x, n = 3, nsim = 5000)

# Expected hits (exact)
En <- 3 * x / sum(x)

# In WOR case: diagonal approximates E(n_k)
diag(pairexp)  # Should be close to En
#> [1] 0.2938 0.6096 0.4508 0.7508 0.8950

# Covariance structure for variance estimation
En_outer <- outer(En, En)
Cov_nn <- pairexp - En_outer  # E(n_i n_j) - E(n_i)E(n_j)