Introduction to turbulence/Statistical analysis/Generalization to the estimator of any quantity
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:
where . It is straightforward to show that this estimator is unbiased, and its variability (squared) is given by:
Example: Suppose it is desired to estimate the variability of an estimator for the variance based on a finite number of samples as:
(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).
Let in equation 2.55 so that , and . Then: