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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

2.2 Direct approach

We directly the fractal nature is to rescale the whole function in abscissa and ordinate and copying it a number of times obtaining a function that is again a distribution function. Consider

the space of distribution functions on [0, 1], then

is a complete metric space, where

Let

be fixed and let:

i)

    with

    and

ii)

    are increasing and continuous;

iii)

iv)

v)

On

we define an operator in the following way (see Iacus and La Torre, 2001):

(4)

open fullsize image

where

From now on we assume that wi are affine maps of the form wi(x) = six + ai, with 0 < si< 1 and

.

Remark that the new distribution function TF is union of distorted copies of F; this is the fractal nature of the operator.

(5)

where

is fixed and:

i)

ii)

iii)

iv)

v)

We limit the treatise to affine maps wi as in Forte and Vrscay (1995), but the general case of increasing and continuous maps can be treated as well (see cited reference of the authors). From now on, we consider the sets of maps wi and parameters δi as given, thus the operator depends only on the probabilities pi and we denote it by Tp.

Stefano M. Iacus, Davide La Torre

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