Rate of Convergence of Estimators Based on Sample Mean

Abstract

The rate of convergence of the point estimators for the parameter based on the sample mean of i.i.d. random vectors is obtained. The result is used to prove the Bahadur's efficiency of the regular best asymptotically normal estimator when the underlying distribution is in the exponential family. An example is given to show that if the distribution is not in the exponential family, then the regular best asymptotically normal estimator is not necessarily efficient in Bahadur's sense.