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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

3 Fourier analysis results

The results presented in this section, taken from Forte and Vrscay (1998) Sec. 6, are rather straight forward to prove but it is essential to recall them since we will use these in density estimation later on.

Given a measure

the Fourier transform


is the complex
space, is defined by the relation

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Table 1: Approximation results for the di erent N-maps IFS (w, p) for the targe distribution function F(x) = x2(3−2x) as in Forte and Vrscay (1995). N = number of maps used, AMSE = average MSE, max p is the contractivity constant of TN in F([0, 1]), s is the contractivity constant of M in M([0, 1]). N' the number of non null probabilities. For the rest of the notation see text.

with the well known properties

We denote by FT(X) the set of all FT's associated to the measures in M(X). Given two elements


FT(X) the following metric can be defined

and the above integral is always finite (see cited paper). With this metric (FT(X), dFT ) is a complete metric space. Given an N-maps affine IFS(w, p) and its Markov operator M it is possibile to define a new linear operator B : FT(X) → FT(X) as follows


 is the FT of a measure µ and is the FT of ν = Mµ.

Stefano M. Iacus, Davide La Torre

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