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The usual estimator for the variance is the corrected sample variance :. If we define. Values of MSE may be used for comparative purposes. Two or more statistical models may be compared using their MSEs—as a measure of how well they explain a given set of observations: An unbiased estimator estimated from a statistical model with the smallest variance among all unbiased estimators is the best unbiased estimator or MVUE Minimum Variance Unbiased Estimator.
Both linear regression techniques such as analysis of variance estimate the MSE as part of the analysis and use the estimated MSE to determine the statistical significance of the factors or predictors under study.
The goal of experimental design is to construct experiments in such a way that when the observations are analyzed, the MSE is close to zero relative to the magnitude of at least one of the estimated treatment effects.
In one-way analysis of variance , MSE can be calculated by the division of the sum of squared errors and the degree of freedom. Also, the f-value is the ratio of the mean squared treatment and the MSE.
MSE is also used in several stepwise regression techniques as part of the determination as to how many predictors from a candidate set to include in a model for a given set of observations.
Squared error loss is one of the most widely used loss functions in statistics [ citation needed ] , though its widespread use stems more from mathematical convenience than considerations of actual loss in applications.
Carl Friedrich Gauss , who introduced the use of mean squared error, was aware of its arbitrariness and was in agreement with objections to it on these grounds.
The use of mean squared error without question has been criticized by the decision theorist James Berger. Mean squared error is the negative of the expected value of one specific utility function , the quadratic utility function, which may not be the appropriate utility function to use under a given set of circumstances.
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