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

(Difference between revisions)
 Revision as of 11:26, 10 June 2006 (view source)Michail (Talk | contribs)← Older edit Revision as of 11:37, 10 June 2006 (view source)Michail (Talk | contribs) Newer edit → Line 28: Line 28: Let $f=(x-X)^2$ in equation 2.55 so that $F_{N}= var_{N}\left\{ x \right\}$, $\left\langle f \right\rangle = var \left\{ x \right\}$ and $var \left\{f \right\} = var \left\{ \left( x-X \right)^{2} - var \left[ x-X \right] \right\}$. Then: Let $f=(x-X)^2$ in equation 2.55 so that $F_{N}= var_{N}\left\{ x \right\}$, $\left\langle f \right\rangle = var \left\{ x \right\}$ and $var \left\{f \right\} = var \left\{ \left( x-X \right)^{2} - var \left[ x-X \right] \right\}$. Then: + +
+ :$+ \epsilon^{2}_{F_{N}}= \frac{1}{N} \frac{var \left\{ \left( x-X \right)^{2} - var \left[x \right] \right\} }{ \left( var \left\{ x \right\} \right)^{2} +$ + (2)
+ + This is easiest to understand if we first expand only the numerator to oblain: + +
+ :$+ var \left\{ \left( x- X \right)^{2} - var\left[x \right] \right\} = \left\langle \left( x- X \right)^{4} \right\rangle - \left[ var \left\{ x \right\} \right]^2 +$ + (2)
+ + Thus + +
+ :$+ \epsilon^{2}_{var_{N}} = \frac{\left\langle \left( x- X \right)^4 \right\rangle}{\left[ var \left\{ x \right\} \right]^2 } - 1 +$ + (2)

## Revision as of 11:37, 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)

(Note that this estimator is not really very useful since it presumes that the mean value, $X$, is known, whereas in fact usually only $X_{N}$ is obtainable).

Let $f=(x-X)^2$ in equation 2.55 so that $F_{N}= var_{N}\left\{ x \right\}$, $\left\langle f \right\rangle = var \left\{ x \right\}$ and $var \left\{f \right\} = var \left\{ \left( x-X \right)^{2} - var \left[ x-X \right] \right\}$. Then:

 Failed to parse (syntax error): \epsilon^{2}_{F_{N}}= \frac{1}{N} \frac{var \left\{ \left( x-X \right)^{2} - var \left[x \right] \right\} }{ \left( var \left\{ x \right\} \right)^{2} (2)

This is easiest to understand if we first expand only the numerator to oblain:

 $var \left\{ \left( x- X \right)^{2} - var\left[x \right] \right\} = \left\langle \left( x- X \right)^{4} \right\rangle - \left[ var \left\{ x \right\} \right]^2$ (2)

Thus

 $\epsilon^{2}_{var_{N}} = \frac{\left\langle \left( x- X \right)^4 \right\rangle}{\left[ var \left\{ x \right\} \right]^2 } - 1$ (2)