Approximation of Bayes estimates by sums of independent random variables (Q1819851)

Let \((X,{\mathcal B},\nu)\) be a measure space and let \((\xi_ 1,...,\xi_ n)\) be a sample consisting of i.i.d. random variables with common distribution density \(f(x,\theta)\) (with respect to \(\nu)\) where \(\theta\) is an unknown parameter with values in a convex bounded set \(\Theta \subset {\mathbb{R}}^ k\), \(k\geq 1\); \(\theta_ 0\) is the true value of \(\theta\). Denote \(\ell (x,\theta)=\ln f(x,\theta)\), \({\mathcal I}(\theta)\) is the Fisher information matrix, \(\hat t_ n\) the Bayes estimate of \(\theta\) with respect to a prior density \(q(\theta)>0\). Under some conditions it is proved that there exists an \(\epsilon >0\) such that with \(P_{\theta_ 0}\)-probability 1 \[ \sqrt{n}{\mathcal I}(\theta_ 0)(\hat t_ n-\theta_ 0)=(1/\sqrt{n})\sum^{n}_{j=1}(\partial /\partial \theta)\ell (\xi_ j,\theta_ 0)+o(n^{-\epsilon})\quad as\quad n\to \infty. \] This theorem extends a well-known result due to \textit{I. A. Ibragimow} and \textit{R. Z. Khas'minskij} [Dokl. Akad. Nauk SSSR 210, 1273- 1276 (1973; Zbl 0303.62027) and ''Asymptotic theory of statistical estimation.'' (1979; Zbl 0467.62025)] to the case when the distribution density vanishes at some points.