## On fractal distribution function estimation and applications

# Theorem 13

Let

Then

is asymptotically efficient under the hypotheses of Theorem 12. Proof. Note that *R*_{0}(*F*_{0}) is a lower bound on the asymptotic risk of

by Theorem 12. Moreover,

open full size image

by Chauchy-Schwartz inequality applied to

as

we have the result. Note that α > 1 is not admissible as at most N_{n} = n quantiles are of statistical interest.

Stefano M. Iacus, Davide La Torre

**Next: ** 4.2 Characteristic function and Fourier density estimation

**Summary: **Index