Estimator of the variance of the Horvitz-Thompson estimator. It is based on the variance estimator of the conditional Poisson sampling design. See Tillé (2020, Chapter 5) for more informations.
varHAJ(y, pik, s)A number, the variance.
vector of size \(n\) that represent the variable of interest.
vector of the inclusion probabilities. The length should be equal to \(n\).
index vector of size \(n\) with elements equal to the selected units.
The function computes the following quantity :
$$v_{HAJ}(\widehat{Y}_{HT}) = \frac{n}{n-1} \sum_{k\in S} (1-\pi_k)\left( \frac{y_k}{\pi_k}-\frac{ \sum_{l\in S} (1-\pi_k)/\pi_k }{\sum_{l\in S} (1-\pi_k) } \right)^2 .$$
This estimator is well-defined for maximum entropy sampling design and use only inclusion probabilities of order one.
Tillé, Y. (2020). Sampling and estimation from finite populations. New York: Wiley