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Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations

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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, X_{N}. For N independent realizations x_{n}, n=1,2,...,N where N is finite, X_{N} is given by:

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