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Theorem 12 (Gill and Levit, 1995)

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

If *F* is a rich family, then for any estimator F_{n} of *F*,

where V ↓ F_{0} denotes the limit in the net of shrinking neighborhoods (with respect to the variation distance) of F_{0} and

The above theorem states that, for any fixed *F*_{0}, it is impossible to do better that *R*_{0}(*F*_{0}) when we try to estimate *F*_{0}. The empirical distribution function

is such

for all n and so it is asymptotically efficient in the sense above mentioned. The result follows from the continuity of *R*_{n} in the variation distance topology (see Gill and Levit, 1995). It is almost trivial to show that also the quantile-based IFS estimator is asymptotically efficient in the sense of the minimax theorem, the only condition to impose is on the number of maps

*N*_{n} as in the LIL result.

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