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Statistical Methods for Fuzzy Data by Reinhard Viertl

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10.1 Observation space and sample space

For a classical random variable X the set Mx of all possible values for X is called observation space, i.e.

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Let X1,…, Xn be a random sample of X, i.e. X1, … , Xn are identically and independently distributed random variables having the same distribution as X. Then the set of all possible values which (X1, … , Xn) can take is the Cartesian product of n copies of the observation space, i.e.

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called sample space.

If x1, …, xn is an observed sample from X then the observations xi are elements of the observation space Mx, i.e. xiMx for all i = 1(1)n.

Therefore the vector (x1, … , xn), which is the combined sample, is an element of the sample space, i.e. (x1, … , xn) ∈ Mxn. For precise observations this needs no further explanation. Moreover so-called statistics are functions of the sample, i.e. s(x1, … , xn), with s : MxnN, where N is a suitable set. The function S = s(X1, … , Xn) is also called statistic.

For fuzzy data the situation is different, because of the fact that the observations are fuzzy numbers xi* with characterizing functions ξi(·). An observed fuzzy sample x1*, … , xn* consists of n fuzzy numbers. But these fuzzy numbers do not form a fuzzy element of the sample space Mxn. So (x1*, … , xn*) is not a suitable object to generalize ...

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