Asymptotical behaviour of some statistical estimators. II. Limiting theorems for the a posteriory density and Bayesian estimators

I. A. Ibragimov   ; R. Z. Khas'minskii

Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 1, pp. 78-93

Résumé

In the second part of the paper we use propositions, methods and results of the first part appeared in the previous issue of this journal. Under conditions I–IV of § 1, we prove theorems about behaviour of the a posteriory density (similar to the well-known Le Cam's results [2]), Bayesian estimators $t_n^{(a)}$ for the risk function $\|\theta\|^a$, Pitman's estimators of the location parameter etc. We prove, for example, that the estimators $t_n^{(a)}$, for different $a\ge1$, are equivalent in the sense that $$ \mathbf E\{\sqrt n\bigl|t_n^{(a_1)}-t_n^{(a_2)}\bigr|\}^p\underset{n\to\infty}\longrightarrow0\quad(p>0). $$