CHANGE LANGUAGE |
Home >
Doc >
On fractal distribution function estimation and applications >
2.3 The choice of affine maps

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 we are mostly concerned with estimation, we briefly discuss the problem of choosing the maps. In Forte and Vrscay (1995) the following two sets of wavelet-type maps are proposed. Fixed and index

define

and

Then set

respectively. To choose the maps, consider the natural ordering of the maps ω_{ij} and operate as follows

and

respectively. Our quantile based maps are of the following type

where

and

are N + 1 equally spaced points on [0, 1]. For each given sets of maps w (W_{1}, W_{1} and W_{q}) di erent **p**'s will be solution of (**Q**) (or (**P**)). Whether the corresponding fixed point is closer to a given *F* in the three cases is not always clear. As an example, in Table 1 we show the relative performance of the approximation based on the quantity

(that is an approximation of the collage distance), on the sup-norm d1 and on the average mean square error, AMSE. We also report the contractivity constant in both the space *M*([0, 1]) and the space *F*([0, 1]). Recall that the collage theorem for the moments establishes that, if **g** is the vector of moments of a the target measure µ (of a distribution function *F*) and is the moment vector of the invariant measure ¯µ* _{N}* of the IFS (

Table 1 shows that, at least in this classical example of the **IFS** literature, for a fixed number of maps *N*, *T _{N}* is a better approximator than M relatively to the sup-norm and the AMSE while the contrary is true in terms of the approximate collage distance Δ

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