Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations
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Revision as of 06:56, 7 June 2006
Estimators for averaged quantities
Since there can never an infinite number of realizations from which ensemble averages (and probability densities) can be computed, it is essential to ask: How many realizations are enough? The answer to this question must be sought by looking at the statistical properties of estimators based on a finite number of realization. There are two questions which must be answered. The first one is:
- Is the expected value (or mean value) of the estimator equal to the true ensemble mean? Or in other words, is yje estimator unbiased?
The second question is
- Does the difference between the and that of the true mean decrease as the number of realizations increases? Or in other words, does the estimator converge in a statistical sense (or converge in probability). Figure 2.9 illustrates the problems which can arise.
Bias and convergence of estimators
A procedure for answering these questions will be illustrated by considerind a simple estimator for the mean, the arithmetic mean considered above, . For independent realizations where is finite, is given by:
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