Strong convergence of distributions of estimators
Let the parameter space \(\theta\) be an open subset of R k and \(\{T_ n\}^ a \)sequence of estimators of \(\vartheta\in \theta\). Assuming that the locally asymptotically normal condition is satisfied at \(\vartheta_ 0\in \theta\), the author obtained a sufficient condition which assures that for all \(\alpha >0\lim_{n\to \infty}\sup_{| h| \leq \alpha}\sup_{| f| \leq 1}| E[f(T\quad *\quad_ n-h| Q_{h,n})]-\int f(x)L(dx)| =0\) where \(T\) \(*_ n=\delta_ n^{- 1}(T_ n-\vartheta_ 0)\), \(\delta_ n\) is some positive definite matrix, \(h\in R\) k, f is a Borel measurable function and L denotes the limit distribution of \(T\) \(*_ n\) under \(Q_{0,n}.\) This convergence is weaker than the convergence of densities but sufficient for many statistical purposes. The method of proof used in this paper improves \textit{D. D. Boos}' method, Ann. Stat. 13, 423-427 (1985; Zbl 0567.62012) and seems to be applicable to establish such convergence of densities.
scientific article; zbMATH DE number 4039013