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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

Corollary 6

As a corollary of the collage Theorem 4 we can anwser to the question: how many quantiles are needed to approximate a distribution function with a given precision, say ε ? The answer is: take the first integer N such that

This value of N is in fact the one that guarantees that the sup-norm distance between the true F and the fixed point

of TF is less than ε. In general, this distance could be considerably smaller as shown in Table 1. Proof. To estimate d1(TFF, F) we can slipt the interval [0,1] as

In each of the interval [qi, qi+1) the distance between TF F ( x ) and F ( x ) is most 1/N. So, by the collage theorem, we have

since

So, given ε > 0, it is suffcient to choose

To investigate the asymptotic behaviour of TF it is worth to show the relation between this functional on the space of distribution function and the one proposed by Forte and Vrscay (1995) on the space of measures. Assuming that for any

we have

and

then TF can be rewritten as

letting

Stefano M. Iacus, Davide La Torre

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