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Statistical Methods for Fuzzy Data by Reinhard Viertl

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11.2 Sample moments

Without any parametric assumptions sample moments can be used to estimate moments of probability distributions.

The first sample moment t1(x1,…, xn) is the sample mean

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which is used to estimate the first moment (i.e. the expectation) of a stochastic quantity.

The second centered sample moment t2(x1,…, xn) is the sample variance

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which is used to estimate the variance of a stochastic quantity.

For k > 2 so-called higher sample moments mk are defined by

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In the case of fuzzy data x1*,…, xn* sample moments yield fuzzy values whose characterizing functions are given as in Section 11.1.

In the case where all fuzzy data are fuzzy intervals the δ-cuts of the sample moments are obtained by application of Theorem 3.1 since the sample moments are continuous functions of the data x1,…, xn:

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For the sample mean n* = m1* this reduces to the formula in Remark 11.2.

For the sample variance and its square root, called sample dispersion Sn*, the calculation is more complicated.

For details, ...

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