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On fractal distribution function estimation and applications


Theoretical background for affine IFS

2.1 Minimization approach

Theorem 1

Theorem 2 (Collage theorem)

Theorem 3 (Forte and Vrscay, 1995)

2.2 Direct approach

Theorem 4 (Iacus and La Torre, 2001).

Theorem 5 (Iacus and La Torre, 2001)

Corollary 6

Theorem 7

2.3 The choice of affine maps

3 Fourier analysis results

Theorem 8 (Forte and Vrscay, 1998)

Theorem 9 (Collage Theorem for FT, (Forte and Vrscay, 1998))

4 Statistical applications

4.1 Asymptotic results for the quantile-based IFS estimator

Theorem 10

Theorem 11

Theorem 12 (Gill and Levit, 1995)

Theorem 13

4.2 Characteristic function and Fourier density estimation

5 Monte Carlo analysis

5.1 Applications to survival analysis

Final remarks


On fractal distribution function estimation and applications

Theorem 2 - Collage theorem

Let (Y, dY ) be a complete metric space. Given an

suppose that there exists a contractive map f on Y with contractivity factor 0 ≤ c < 1 such that

If is the fixed point of


So if one wishes to approximate a function y with the fixed point of an unknown contractive map f, it is only needed to solve the inverse problem of finding f which minimizes the collage distance dY (y,f(y)). The main result in Forte and Vrscay that we will use to build one of the IFSs estimators is that the inverse problem can be reduced to minimize a suitable quadratic form in terms of the pi given a set of affine maps wi and the sequence of moments gk of the target measure. Let

be the simplex of probabilities. Let w = (w1,w2, . . . ,wn), N = 1, 2, . . . be subsets of W = {w1,w2, . . .} the infinite set of affine contractive maps on X = [0, 1] and let g the set of the moments of any order of

Denote by M the Markov operator of the N-maps IFS (w, p) and by

, with associated moment vector of any order hN. The collage distance between the moment vector of

is a continuous function and attains an absolute minimum value Δ min on ΠN

Stefano M. Iacus, Davide La Torre

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