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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 13

Let

Then

is asymptotically efficient under the hypotheses of Theorem 12. Proof. Note that R0(F0) is a lower bound on the asymptotic risk of

by Theorem 12. Moreover,

open full size image

by Chauchy-Schwartz inequality applied to

as

we have the result. Note that α > 1 is not admissible as at most Nn = n quantiles are of statistical interest.

Stefano M. Iacus, Davide La Torre

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