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Theorem 5 (Iacus and La Torre, 2001)

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

Choose

and

such that

for some

where

is the fixed point of T_{p} on

and c is the contractivity constant of T_{p}. Moreover, the function

is convex.

Thus, the following constrained optimization problem:

(**P**)

can always be solved at least numerically. Another way of choosing the form of T_{p} is the direct approach, that is the following. Choose n = N +1 points on [0, 1], (x_{1}, . . . , x_{n}), and assume that 0 = x_{1} < x_{2} < · · · < x_{ n−1}< x_{n} = 1. The proposed functional is the following

i = 1, . . . , n−1, where u is any member in the space

Notice that *T _{F}* is a particular case of

and

This is a contraction and, at each iteration, *T _{F}* passes exactly through the points F(x

This form of the estimator proposes an intuitive (possibily) good candidate for distribution function estimation. Note that we overcome the problem of moment matching as we don't even need the existence of the moments.

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