Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity
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(Difference between revisions)
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:<math> | :<math> | ||
F_{N}\equiv\frac{1}{N}\sum^{N}_{n=1}f_{n} | F_{N}\equiv\frac{1}{N}\sum^{N}_{n=1}f_{n} | ||
| + | </math> | ||
| + | </td><td width="5%">(2)</td></tr></table> | ||
| + | |||
| + | where <math>f_{n}\equiv f(x_{n})</math>. It is straightforward to show that this estimator is unbiased, and its variability (squared) is given by: | ||
| + | |||
| + | <table width="100%"><tr><td> | ||
| + | :<math> | ||
| + | \epsilon^{2}_{F_{N}}= \frac{1}{N} \frac{var \left\{f \left( x \right) \right\}}{\left\langle f \left( x \right) \right\rangle^{2} } | ||
| + | </math> | ||
| + | </td><td width="5%">(2)</td></tr></table> | ||
| + | |||
| + | '''Example:''' Suppose it is desired to estimate the variability of an estimator for the variance based on a finite number of samples as: | ||
| + | |||
| + | <table width="100%"><tr><td> | ||
| + | :<math> | ||
| + | var_{N} \left\{x \right\} \equiv \frac{1}{N} \sum^{N}_{n=1} \left( x_{n} - X \right)^{2} | ||
</math> | </math> | ||
</td><td width="5%">(2)</td></tr></table> | </td><td width="5%">(2)</td></tr></table> | ||
Revision as of 11:14, 10 June 2006
Similar relations can be formed for the estimator of any function of the random variable say 
. For example, an estimator for the average of 
based on 
realizations is given by:
|
| (2) |
where 
. It is straightforward to show that this estimator is unbiased, and its variability (squared) is given by:
|
| (2) |
Example: Suppose it is desired to estimate the variability of an estimator for the variance based on a finite number of samples as:
|
| (2) |
