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Final remarks

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

We haw shown how it is relatively powerful to adopts IFS technique in distribution function estimation and related quanties (density and Fourier transform). There are several open issues about the estimators themselves.

The main open problem is about a better characterization of the fixed points of the IFS in order to establish non asymptotic properties for the estimators. The second, and commonly not discussed in the IFS literature, is the problem of choosing the maps w.

There recently appeared some papers that discuss the relationships of some class of IFS and wavelets analysis as well as some papers on local IFS (possible candidates to density functions approximators) but the results there in are not directly useful to statistics.

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