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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 11

Let F be continuous and

Then

has the Chung-Smirnov property.

Proof. In fact,

by hypotheses.

We can also establish the local asymptotic minimax optimality of our estimator when F is in a rich family (in the sense of Levit, 1978 and Millar 1979, see as well Gill and Levit, 1995, Section 6) of distribution functions. For any estimator Fn of the unknown distribution function F we define the integrated mean square error as follows

open full size image

where

is a fixed probability measure on [0,1] and EF is the expectation under the true law F. What follows is the minimax theorem in the version given by Gil and Levit (1995).

Stefano M. Iacus, Davide La Torre

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