Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity
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Let <math>f=(x-X)^2</math> in equation 2.55 so that <math>F_{N}= var_{N}\left\{ x \right\}</math>, <math>\left\langle f \right\rangle = var \left\{ x \right\} </math> and <math>var \left\{f \right\} = var \left\{ \left( x-X \right)^{2} - var \left[ x-X \right] \right\}</math>. Then: | Let <math>f=(x-X)^2</math> in equation 2.55 so that <math>F_{N}= var_{N}\left\{ x \right\}</math>, <math>\left\langle f \right\rangle = var \left\{ x \right\} </math> and <math>var \left\{f \right\} = var \left\{ \left( x-X \right)^{2} - var \left[ x-X \right] \right\}</math>. Then: | ||
+ | |||
+ | <table width="100%"><tr><td> | ||
+ | :<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | </td><td width="5%">(2)</td></tr></table> | ||
+ | |||
+ | This is easiest to understand if we first expand only the numerator to oblain: | ||
+ | |||
+ | <table width="100%"><tr><td> | ||
+ | :<math> | ||
+ | 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 | ||
+ | </math> | ||
+ | </td><td width="5%">(2)</td></tr></table> | ||
+ | |||
+ | Thus | ||
+ | |||
+ | <table width="100%"><tr><td> | ||
+ | :<math> | ||
+ | \epsilon^{2}_{var_{N}} = \frac{\left\langle \left( x- X \right)^4 \right\rangle}{\left[ var \left\{ x \right\} \right]^2 } - 1 | ||
+ | </math> | ||
+ | </td><td width="5%">(2)</td></tr></table> |
Revision as of 11:37, 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) |
(Note that this estimator is not really very useful since it presumes that the mean value, , is known, whereas in fact usually only is obtainable).
Answer
Let in equation 2.55 so that , and . Then:
| (2) |
This is easiest to understand if we first expand only the numerator to oblain:
| (2) |
Thus
| (2) |