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 11
Let F be continuous and
Then
has the Chung-Smirnov property.
Proof. In fact,
by hypotheses.
We can also establish the local asymptotic minimax optimality of our estimator when F is in a rich family (in the sense of Levit, 1978 and Millar 1979, see as well Gill and Levit, 1995, Section 6) of distribution functions. For any estimator Fn of the unknown distribution function F we define the integrated mean square error as follows
where
is a fixed probability measure on [0,1] and EF is the expectation under the true law F. What follows is the minimax theorem in the version given by Gil and Levit (1995).
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