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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

Theorem 12 (Gill and Levit, 1995)

If F is a rich family, then for any estimator Fn of F,

where V ↓ F0 denotes the limit in the net of shrinking neighborhoods (with respect to the variation distance) of F0 and

The above theorem states that, for any fixed F0, it is impossible to do better that R0(F0) when we try to estimate F0. The empirical distribution function

is such

for all n and so it is asymptotically efficient in the sense above mentioned. The result follows from the continuity of Rn in the variation distance topology (see Gill and Levit, 1995). It is almost trivial to show that also the quantile-based IFS estimator is asymptotically efficient in the sense of the minimax theorem, the only condition to impose is on the number of maps

Nn as in the LIL result.

Stefano M. Iacus, Davide La Torre

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