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**Extra resources for A nonparametric approach to the analysis of longitudinal data via a set of level crossing problems w**

**Example text**

The standard deviation √ of µ(ψ) as computed from the diﬀerent samples tends towards σ(Ψ )/ N . Thus, if we compute the sample mean from a given sample, we can expect that the error √and given √ in the computed sample mean is normally distributed by σ(Ψ )/ N . Importantly, the error decreases proportional to 1/ N . The amazing and counter-intuitive property of the central limit theorem is that no matter what the shape of the original distribution, the sampling distribution of the mean approaches a normal distribution.

The subscript (3) on Mi deﬁnes the variable the functional is operating on, thus, the integration variable is x3 . Multiplying this equation with δψ(x1 ) and integrating in x1 now gives D δψ(x1 )Mi(3) [δ(x1 − x3 )]dx1 = D δψ(x1 )δ(x1 − xi )dx1 = Mi(1) [δψ(x1 )] = Mi(3) [δψ(x3 )]. 33) where in the last line, we changed the dummy variable for the integration to x3 . 31). 34) f = Mi(2) [Cψψ (x1 , x2 )]. e. the measurement term, is constant in the integration with respect to x2 . 34) are veriﬁed for i = 1, .

Further, we have deﬁned the vector of measurement errors ∈ M . The measurement errors may be a composite of errors introduced when measuring the variable and additional representation errors introduced when constructing the measurement functional. This will be discussed in more detail in the following chapters. 24) where xi is the measurement location, δ(x−xi ) is the Dirac delta function, and the subscript i denotes the component i of the measurement functional. g. 24). 2 Extension to spatial dimensions 17 The actual values of the errors pf (x) and are not known.