On a converse to Scheffé's theorem
Let \(\{G_ n\}\) be a sequence of absolutely continuous distributions on \({\mathbb{R}}^ k\) and let \(\{g_ n\}\) be the corresponding sequence of densities. The main theorem says: The following two statements are equivalent: (1) \(\{g_ n\}\) is a.e.c. and bounded and \(G_ n\Rightarrow G.\) (2) \(g_ n\to g\) pointwise, uniformly in compacts of \({\mathbb{R}}^ k\) when g is the continuous density of the distribution G. a.e.c. stands for ''asymptotically equicontinuous'' what is defined as follows: for any \(x\in {\mathbb{R}}^ k\) and \(\epsilon >0\) there exist \(\delta =\delta (x,\epsilon)\) and \(n_ 0=n_ 0(x,\epsilon)\) such that whenever \(| y-x| \leq \delta\) then \(| g_ n(y)-g_ n(x)| <\epsilon\) for all \(n\geq n_ 0\) where \(| y-x|\) is the maximum absolute coordinate. This and the next similar result is an extension of an earlier result of \textit{D. D. Boos} [ibid. 13, 423-427 (1985; Zbl 0567.62012)]. An application to likelihood functions is also given.
uniform approximation of likelihood functions
scientific article; zbMATH DE number 3978119