## 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) **

**ii) **

**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 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

**Next: ** Theorem 4 (Iacus and La Torre, 2001).

**Summary: **Index