If F is a rich family, then for any estimator Fn of F,
where V ↓ F0 denotes the limit in the net of shrinking neighborhoods (with respect to the variation distance) of F0 and
The above theorem states that, for any fixed F0, it is impossible to do better that R0(F0) when we try to estimate F0. The empirical distribution function
for all n and so it is asymptotically efficient in the sense above mentioned. The result follows from the continuity of Rn 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
Nn as in the LIL result.
Stefano M. Iacus, Davide La Torre
Next: Theorem 13.