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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

4.1 Asymptotic results for the quantile-based IFS estimator

Asymptotic properties of the fixed points of both

and

derive as a natural consequence, by the properties of the empirical moments and quantiles. So, one can expect that, for a fixed number of N maps, the fixed point of

is a consistent estimator of the fixed point of M as the sample size increases and that the fixed point of

converges to the fixed point of TN as well. But if we let N varying with the sample size n we can have much more, at least from

The fixed point

of the above operator,

satisfies

for real x. The following (Glivenko-Cantelli) theorem states that

has the same properties of an admissible perturbation of the e.d.f in the sense of Winter (see Winter 1973, 1979 and Yukish, 1989). Let us denote by Nn the number of maps and coefficients in the IFS so to put in evidence the dependency of the sample size n.

Stefano M. Iacus, Davide La Torre

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