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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 3 (Forte and Vrscay, 1995).

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 hN = Ag. In particular





The series above are convergent as 0 ≤ Ani ≤ 1 and the minimum can be found by minimizing the quadratic form on the simplex ΠN. 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

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