A New Estimator of the Variance Based on Minimizing Mean Squared Error

Abstract

In 2005, Yatracos constructed the estimator S ² ₂ = c ₂ S ², c ₂ = (n + 2)(n − 1)[n(n + 1)]^{− 1}, of the variance, which has smaller mean squared error (MSE) than the unbiased estimator S ². In this work, the estimator S ² ₁ = c ₁ S ², c ₁ = n(n − 1)[n(n − 1) + 2]^{− 1}, is constructed and is shown to have the following properties: (a) it has smaller MSE than S ² ₂, and (b) it cannot be improved in terms of MSE by an estimator of the form cS ², c > 0. The method of construction is based on Stein’s classical idea brought forward in 1964, is very simple, and may be taught even in an undergraduate class. Also, all the estimators of the form cS ², c > 0, with smaller MSE than S ² as well as all those that have the property (b) are found. In contrast to S ², the method of moments estimator is among the latter estimators.

Keywords:

• Admissibility
• Stein’s method
• Variance estimation