Hansen-Hurwitz estimator
The Hansen-Hurwitz estimator
The Hansen-Hurwitz estimator gives the framework for all unequal probability sampling with replacement (Hansen and Hurwitz, 1943[1]). “With replacement” means that the selection probabilities are the same for all draws; if selected elements would not be replaced (put back to the population), the selection probabilities would change after each draw for the remaining elements.
Suppose that a sample of size n is drawn with replacement and that on each draw the probability of selecting the i-th unit of the population is \(p_i\).
Then the Hansen-Hurwitz estimator of the population total is
\[\hat \tau = \frac {1}{n} \sum_{i=1}^n \frac {y_i}{p_i}\]
Here, each observation \(y_i\) is weighted by the inverse of its selection probability \(p_i\).
The parametric variance of the total is
\[var (\hat \tau) = \frac {1}{n} \sum_{i=1}^N p_i \left (\frac {y_i}{p_i} - \tau \right )^2\]
which is unbiasedly estimated from a sample size n from
\[v\hat ar (\hat \tau) = \frac {1}{n} \frac {\sum_{i=1}^n \left (\frac {y_i}{p_i} - \tau \right )^2}{n-1}\]
Hansen-Hurwitz estimator examples: 4 application examples
References
- ↑ Hansen MM and WN Hurwitz. 1943. On the theory of sampling from finite populations. Annals of Mathematical Statistics 14:333-362.
