## 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 F_{n} of the unknown distribution function *F* we define the integrated mean square error as follows

open full size image

where

is a fixed probability measure on [0,1] and E_{F} 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

**Next: ** Theorem 12 (Gill and Levit, 1995).

**Summary: **Index