Lower Bounds on Variance of Estimators Past, Present and Future

The approach to obtain uniformly minimum variance unbiased estimator (UMVUE) through the lower bound on the variance of an unbiased estimator involves two problems: 1. To provide a lower bound and 2. To identify the classes of (a) probability distributions (b) estimators and (c) parametric functions for which a particular bound is attained by the variance of the UMVUE. Many authors have provided various solutions to these problems starting from Fre chet (1943), Crame r (1946) and Rao (1945). In this paper we are trying to provide a brief history of lower bounds on the variance of unbiased estimators. It is tried to take an account of little bit of present and possible future work in the field of non-regular families of distributions. Sometimes different inequalities provide different bounds for the variance of estimators of the same parametric function. In such situations, it is natural to compare the bounds. We have made a brief mention of these efforts also. A few important references are provided.