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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 10

Let

be as in (6) with

as

If F is continuous, then

Proof. We can write

open full size image

and the first term can always be estimated by 1/Nn while the second one converges to 0 almost surely by Glivenko-Cantelli theorem. We can also establish a result of LIL-type. Recall that (Winter, 1979) an estimator Fn of F is said to have the Chung-Smirnov property if

with probability 1.

Stefano M. Iacus, Davide La Torre

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