Hansen-Hurwitz estimator
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| + | The Hansen-Hurwitz estimator gives the framework for all [[Sampling with unequal selection probabilities|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 probability|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>. | |
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| − | Suppose | + | |
Then the Hansen-Hurwitz estimator of the population total is | Then the Hansen-Hurwitz estimator of the population total is | ||
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:<math>\hat \tau = \frac {1}{n} \sum_{i=1}^n \frac {y_i}{p_i}</math> | :<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 | |
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| − | The parametric variance of the total is | + | |
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:<math>var (\hat \tau) = \frac {1}{n} \sum_{i=1}^N p_i \left (\frac {y_i}{p_i} - \tau \right )^2</math> | :<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 [[bias|unbiasedly]] estimated from a sample size ''n'' from | |
| − | which is unbiasedly estimated from a sample size ''n'' from | + | |
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:<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> | :<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> | ||
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|text=4 application examples | |text=4 application examples | ||
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==References== | ==References== | ||
Latest revision as of 13:20, 26 October 2013
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
[edit] References
- ↑ Hansen MM and WN Hurwitz. 1943. On the theory of sampling from finite populations. Annals of Mathematical Statistics 14:333-362.
