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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

2 Theoretical background for affine IFSs

In this section we recall some of the results from Forte and Vrscay (1995) and Iacus and La Torre (2001) concerning the IFSs setup on the the space of distribution function. LetM(X) be the set of probability measures on B(X), the -algebra of Borel subsets of X where (X, d) is a compact metric space (in our case will be X = [0, 1] and d the Euclidean metric.) In the IFS literature the following Hutchinson metric plays a crucial role


thus (M(X), dH) is a complete metric space (see Hutchinson, 1981).

As usual, we denote by (w, p) an N-maps contractive IFS on X with probabilities or simply an N-maps IFS, that is, a set of N affine contractions maps, w = (w1,w2, . . . ,wn),

with associated probabilities p = (p1, p1, . . . , pn),


The IFS has a contractivity factor defined as

Consider the following (usually called Markov) operator M :

defined as


where wi1
i is the inverse function of wi and ffi stands for the composition. In Hutchinson (1981) it was shown that M is a contraction mapping on (M(X), dH): for all

Thus, there exists a unique measure

the invariant measure of the IFS, such that

by Banach theorem.

Stefano M. Iacus, Davide La Torre

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