Some Aspects of Estimators for Variance of Normally Distributed Data

Normally distributed data arises in various contexts and often one is interested in estimating its variance. The authors limit themselves in this chapter to the class of estimators that are (positive) multiples of sample variances. Two important qualities of estimators are bias and variance, which respectively capture the estimator’s accuracy and precision. Apart from the two classical estimators for variance, they also consider the one that minimizes the Mean Square Error (MSE) and another that minimizes the maximum of the square of the bias and variance, the minmax estimator. This minmax estimator can be identified as a fixed point of a suitable function. For moderate to large sample sizes, the authors argue that all these estimators have the same order of MSE. However, they differ in the contribution of bias to their MSE. The authors also consider their Pareto efficiency in squared bias versus variance space. All the above estimators are non-dominated (i.e., they lie on the Pareto frontier).