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
Theorem 10
Let
be as in (6) with
as
If F is continuous, then
Proof. We can write
and the first term can always be estimated by 1/Nn while the second one converges to 0 almost surely by Glivenko-Cantelli theorem. We can also establish a result of LIL-type. Recall that (Winter, 1979) an estimator Fn of F is said to have the Chung-Smirnov property if
with probability 1.
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