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

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

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

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

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:

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.

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