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
On fractal distribution function estimation and applications
3 Fourier analysis results
The results presented in this section, taken from Forte and Vrscay (1998) Sec. 6, are rather straight forward to prove but it is essential to recall them since we will use these in density estimation later on.
Given a measure
the Fourier transform
where
is the complex
space, is defined by the relation
Table 1: Approximation results for the di erent N-maps IFS (w, p) for the targe distribution function F(x) = x2(3−2x) as in Forte and Vrscay (1995). N = number of maps used, AMSE = average MSE, max p is the contractivity constant of TN in F([0, 1]), s is the contractivity constant of M in M([0, 1]). N' the number of non null probabilities. For the rest of the notation see text.
with the well known properties
We denote by FT(X) the set of all FT's associated to the measures in M(X). Given two elements
and
FT(X) the following metric can be defined
and the above integral is always finite (see cited paper). With this metric (FT(X), dFT ) is a complete metric space. Given an N-maps affine IFS(w, p) and its Markov operator M it is possibile to define a new linear operator B : FT(X) → FT(X) as follows
where
is the FT of a measure µ and is the FT of ν = Mµ.
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