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Corollary 6

On fractal distribution function estimation and applications

Theoretical background for affine IFS

Theorem 3 (Forte and Vrscay, 1995)

Theorem 4 (Iacus and La Torre, 2001).

Theorem 5 (Iacus and La Torre, 2001)

Theorem 8 (Forte and Vrscay, 1998)

Theorem 9 (Collage Theorem for FT, (Forte and Vrscay, 1998))

4.1 Asymptotic results for the quantile-based IFS estimator

Theorem 12 (Gill and Levit, 1995)

4.2 Characteristic function and Fourier density estimation

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 *T _{F}* is less than ε. In general, this distance could be considerably smaller as shown in Table 1. Proof. To estimate d1(

In each of the interval [q_{i}, q_{i}+1) the distance between *T _{F} 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 *T _{F}* 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 *T _{F}* can be rewritten as

letting

Stefano M. Iacus, Davide La Torre

In this paper we review some recent results concerning the approximations of distribution functions and measures on [0, 1] based on iterated function systems.

Stefano M. Iacus, Davide La Torre