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==The Hansen-Hurwitz estimator==

The  Hansen-Hurwitz estimator gives the framework for all unequal probability  sampling with replacement (Hansen and Hurwitz, 1943<ref> Hansen  MM and WN Hurwitz. 1943. On the theory of sampling from finite  populations. Annals of Mathematical Statistics 14:333-362.  </ref>). “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  <math>p_i</math>.

Then the Hansen-Hurwitz estimator of the population total is


:<math>\hat \tau = \frac {1}{n} \sum_{i=1}^n \frac {y_i}{p_i}</math>


Here,  each observation <math>y_i</math> is weighted by the  inverse of its selection probability <math>p_i</math>. 


The parametric variance of the total is


:<math>var (\hat \tau) = \frac {1}{n} \sum_{i=1}^N p_i \left (\frac {y_i}{p_i} - \tau \right )^2</math>


which is unbiasedly estimated from a sample size ''n'' from


:<math>v\hat  ar (\hat \tau) = \frac {1}{n} \frac {\sum_{i=1}^n \left (\frac  {y_i}{p_i} - \tau \right )^2}{n-1}</math>



{{Exercise
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==References==


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[[Category:Sampling design]]