The statistical information contained in additional observations
Within the setting of \textit{L. LeCam's} theory of comparison of experiments [Asymptotic methods in statistical decision theory. (1986; Zbl 0605.62002)] the pseudo-distance \(\Delta\) (\({\mathcal E}^ n,{\mathcal E}^{n+r_ n})\) between the powers of an experiment \({\mathcal E}\) defined by the measures \(P_{\theta}(\theta \in \Theta)\) are studied asymptotically as \(n\to \infty\). If \({\mathcal E}\) admits \(\sqrt{n}\)- constistent estimates, then \(\Delta\) (\({\mathcal E}^ n,{\mathcal E}^{n+r})\leq Cr/n\) for some constant \(C>0\). If \(\Theta \subset {\mathbb{R}}^ k\) and \({\mathcal E}\) can be locally approximated by a homoscedastic Gaussian experiment, then for \(r_ n=o(n)\) \textit{J. Helgeland}'s lower bound [Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 77-100 (1982; Zbl 0479.62006)] can be improved. Helgeland's upper bound turns out to be sharp. Tools for the proofs of the results are LeCam's randomization criterion for the deficiency \(\delta\) (\({\mathcal E},{\mathcal F})\) between experiments \({\mathcal E}\) and \({\mathcal F}\) and its representation in terms of the Bayes risks corresponding to \({\mathcal E}\) and \({\mathcal F}\), respectively.
scientific article; zbMATH DE number 4030689