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Theorem 3 (Forte and Vrscay, 1995)

On fractal distribution function estimation and applications

Theoretical background for affine IFS

Theorem 3 (Forte and Vrscay, 1995)

Theorem 4 (Iacus and La Torre, 2001).

Theorem 5 (Iacus and La Torre, 2001)

Theorem 8 (Forte and Vrscay, 1998)

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

4.1 Asymptotic results for the quantile-based IFS estimator

Theorem 12 (Gill and Levit, 1995)

4.2 Characteristic function and Fourier density estimation

Thus, the collage distance can be made arbitrarily small by choosing a suitable number of maps and probabilities, N^{*}. By the same authors, the inverse problem can be posed as a quadratic programming one in the following notation

Then by (2) there exists a linear operator A : D(X) → D(X) associated to M such that h_{N} = Ag. In particular

where

Thus

(q)

where

The series above are convergent as 0 ≤ A_{ni} ≤ 1 and the minimum can be found by minimizing the quadratic form on the simplex Π^{N}. This is the main result in Forte and Vrscay (1995) that can be used straight forwardly in statistical applications as we propose in Section 4. On the other side Iacus and La Torre (2001) propose a di erent and direct approach to construction on IFSs on the space of distribution function on [0, 1]. Instead of constructing the IFS by matching the moments, the idea there is to have an IFS that has the same values of the target distribution function on a finite number of points.

Stefano M. Iacus, Davide La Torre

In this paper we review some recent results concerning the approximations of distribution functions and measures on [0, 1] based on iterated function systems.

Stefano M. Iacus, Davide La Torre