Theorem 12 (Gill and Levit, 1995)

If F is a rich family, then for any estimator F_n 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

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