Abstract:

Let $X_1, X_2,\ldots, X_n$, be i.i.d. random variables with a density function $f(x,\theta)$ where $\theta\in\Theta\subset R^k$ is an unknown parameter that we are interested in estimating. Following up robustification procedure presented by Huber [10] we shall study one possible approach for using (non-sample) prior information for robust type estimators and prove some asymptotic properties of introduced estimators. We shall show that the Bayes-type estimators and maximum posterior probability estimators are asymptotically equivalent to the order $O_p(n^{-1})$ or $o_p(n^{-1})$, depending on some regularity conditions. Because of this asymptotic relation, one expects that with an appropriate choice of $\rho$ (i. e such as we would use in generating an M-estimator) we can obtain a Bayesian type estimator with good robustness properties. In addition, if $f(x,\theta) = \exp\{-\rho(X_i,\theta)\}$ then these results lead to relations of maximum likelihood and Bayes' estimators.