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Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity

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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}
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</math> 
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</td><td width="5%">(2)</td></tr></table>
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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:
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<table width="100%"><tr><td> 
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:<math>   
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\epsilon^{2}_{F_{N}}= \frac{1}{N} \frac{var \left\{f \left( x \right) \right\}}{\left\langle f \left( x \right) \right\rangle^{2} }
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</math> 
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</td><td width="5%">(2)</td></tr></table>
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'''Example:''' Suppose it is desired to estimate the variability of an estimator for the variance based on a finite number of samples as:
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<table width="100%"><tr><td> 
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:<math>   
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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 f(x). For example, an estimator for the average of f based on N realizations is given by:

    
F_{N}\equiv\frac{1}{N}\sum^{N}_{n=1}f_{n}
(2)

where f_{n}\equiv f(x_{n}). It is straightforward to show that this estimator is unbiased, and its variability (squared) is given by:

    
\epsilon^{2}_{F_{N}}= \frac{1}{N} \frac{var \left\{f \left( x \right) \right\}}{\left\langle f \left( x \right) \right\rangle^{2} }
(2)

Example: Suppose it is desired to estimate the variability of an estimator for the variance based on a finite number of samples as:

    
var_{N} \left\{x \right\} \equiv \frac{1}{N} \sum^{N}_{n=1} \left( x_{n} - X \right)^{2}
(2)
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