CFD Online Logo CFD Online URL
Home > Wiki > Introduction to turbulence/Statistical analysis/Generalization...

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

From CFD-Wiki

Jump to: navigation, search

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:


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} }

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}

(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\}

My wiki