Mean Squared Error

Let

The mean squared error of the estimator refers to its average squared error, i.e.,

Notes:

• The mean squared error (MSE) provides a measure of the quality of an estimator. We generally prefer estimators with small MSE, the smaller the better.
• Estimators with small bias and standard errors have small MSE as can be readily seen from the fact that
  
  
  The alternative expression for MSE given above is an immediate consequence of the definition
  
  
  and the fact that both the estimator and its estimation error have the same variance.
  
• As an example, the MSE of the sample mean derived from a random sample is
  
  
  If based on a random sample from a normal distribution, the MSE of the biased variance estimator may be shown to be given by
  
  
  
• The figure below shows the probability densities for the errors associated with three different unbiased estimators. Here, MSE equals var(e) since all three estimators have zero bias and so the error distribution with the smallest (largest) spread corresponds to the estimator with the smallest (largest) MSE.
  
  
• The next figure shows the probability densities for the errors associated with three different estimators. The error distribution in the middle corresponds to the estimator with the smallest MSE since all three error distributions have the same variance but only the middle one has a zero bias.