The Rate of Convergence of Consistent Point Estimators

TLDR: In this paper, an upper bound for the rate of convergence of consistent estimators based on sample quantities is derived for the case when the underlying distribution is double-exponential and shown to coincide with the classical Pitman asymptotic relative efficiency.

Abstract: The rate at which the probability $P_\theta\{|t_n - \theta| \geqq \varepsilon\}$ of consistent estimator $t_n$ tends to zero is of great importance in large sample theory of point estimation. The main tools available at present for finding the rate are Bernstein-Chernoff-Bahadur's theorem and Sanov's theorem. In this paper, we give two new techniques for finding the rate of convergence of certain consistent estimators. By using these techniques, we have obtained an upper bound for the rate of convergence of consistent estimators based on sample quantities and proved that the sample median is an asymptotically efficient estimator in Bahadur's sense if and only if the underlying distribution is double-exponential. Furthermore, we have proved that the Bahadur asymptotic relative efficiency of sample mean and sample median coincides with the classical Pitman asymptotic relative efficiency.