varHAJ {WaveSampling}

Hajek-Rosen variance estimator

Description

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.

Usage

varHAJ(y, pik, s)

Arguments

y

vector of size n that represent the variable of interest.

pik

vector of the inclusion probabilities. The length should be equal to n.

s

index vector of size n with elements equal to the selected units.

Details

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.

Value

A number, the variance

References

Tillé, Y. (2020). Sampling and estimation from finite populations. New York: Wiley