The statistical information contained in additional observations - Mammen, Enno (The annals of statistics, 1986)
Let EnEn\mathscr{E}^n be a statistical experiment based on nnn i.i.d. observations. We compare EnEn\mathscr{E}^n with En+rnEn+rn\mathscr{E}^{n+r_n}. The gain of information due to the rnrnr_n additional observations is measured by the deficiency distance Δ(En,En+rn)Δ(En,En+rn)Δ (\mathscr{E}^n, \mathscr{E}^{n+r_n}), i.e., the maximum diminution of the risk functions. We show that under general dimensionality conditions Δ(En,En+rn)Δ(En,En+rn)Δ(\mathscr{E}^n, \mathscr{E}^{n+r_n}) is of order rn/nrn/nr_n/n. Further the behavior of ΔΔΔ is studied and compared for asymptotically Gaussian experiments. We show that the information gain increases logarithmically. The Gaussian and the binomial family turn out to be--in some sense--opposite extreme cases, with the increase of information asymptotically minimal in the Gaussian case and maximal in the binomial.