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

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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).

Answer

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:

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

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