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
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
where
Thus
(q) 
where
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
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