MSE < Variance? A pitfall in calculating the mean square error

Abstract

When calculating the mean square error (MSE), it is possible to encounter a situation where the variance of a parameter of interest is larger than its mean square error. In theory, this is impossible because MSE is the sum of variance and bias squared; even when bias is zero, the MSE should be equal to, and not less than, the variance. This short note explains why this is indeed an error with a mathematical proof, demonstrates how this could happen using a small simulation study, and shows how to avoid making such an error in the derivation of the MSE.