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On fractal distribution function estimation and applications

Introduction

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

References

On fractal distribution function estimation and applications

Final remarks

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

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