Hansen-Hurwitz estimator

[−] Forest inventory

[+] Introduction to forest inventory

[+] Forest mensuration

[+] Introduction to sampling

[+] Plot design

[+] Sampling design

[+] Estimation issues

[+] Planning issues

[+] UNFCCC Implications for Forest Monitoring

[+] Forest Inventory Examples

  Forest Inventory Glossary

The Hansen-Hurwitz estimator gives the framework for all unequal probability sampling with replacement (Hansen and Hurwitz, 1943). “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