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

(Difference between revisions)
 Revision as of 11:05, 10 June 2006 (view source)Michail (Talk | contribs)← Older edit Revision as of 11:14, 10 June 2006 (view source)Michail (Talk | contribs) Newer edit → Line 4: Line 4: :$:[itex] F_{N}\equiv\frac{1}{N}\sum^{N}_{n=1}f_{n} F_{N}\equiv\frac{1}{N}\sum^{N}_{n=1}f_{n} +$ +
(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: + +
+ :$+ var_{N} \left\{x \right\} \equiv \frac{1}{N} \sum^{N}_{n=1} \left( x_{n} - X \right)^{2}$ [/itex] (2)
(2)

## 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)