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

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== Estimators for averaged quantities == | == Estimators for averaged quantities == | ||

- | Since there can never an infinite number of realizations | + | 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: |

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+ | * Is the expected value (or mean value) of the estimator equal to the true ensemble mean? Or in other words, is yje estimator ''unbiased?'' | ||

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+ | The second question is | ||

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+ | * 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 == | == Bias and convergence of estimators == |

## Revision as of 06:00, 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.