Abstract: This is an account of the mathematical formulation of asymptotic efficiency in estimation theory from the point of view of the concentration of the estimators around the true parameter value. The purpose is not to propose any new definition of efficiency, but rather to consider the inter-relationships among some rather scattered existing results with the aim of connecting them into a coherent whole. In the process of doing so, some improvements and extensions will also be given. The theory is developed first in the case of a scalar parameter. It is then extended by a simple argument to cover the estimation of a scalar function of a multi or infinite dimensional parameter.
Key words and phrases: Fisher's information bound, concentration probability, regular estimators, locally asymptotically minimax.