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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 1

Set X = [0, 1] and let µ and

with associated moments of any order gk and

Then, the following statements are equivalent

(here C(X) is the space of continuous functions on X). This theorem gives a way to find and appropriate set of maps and probabilities by solving the so called problem of moment matching.

With the solution in hands, given the convergence of the moments, we also have the convergence of the measures and then the stationary measure of M approximates with given precision (in a sense specified by the collage theorem below) the target measure µ (see Barnsley and Demko, 1985). Next theorem, called the collage theorem is a standard result of IFS theory and is a consequence of Banach theorem.

Stefano M. Iacus, Davide La Torre

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