CHANGE LANGUAGE |
Home >
Doc >
On fractal distribution function estimation and applications >
2.2 Direct approach

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

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)

where

From now on we assume that wi are affine maps of the form w_{i}(x) = s_{i}x + a_{i}, with 0 < s_{i}< 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 w_{i} 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 w_{i} and parameters δ_{i} as given, thus the operator depends only on the probabilities pi and we denote it by T_{p}.

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