mean square error

The mean square error of an estimatorMathworldPlanetmath θ^ of a parameter θ in a statistical model is defined as:


From the definition of the varianceMathworldPlanetmath Var[X]=E[X2]-E[X]2, we can express the mean square error in terms of the bias by expanding the right hand side above:


If θ^ is an unbiased estimatorMathworldPlanetmath, then its mean square error is identical to its variance: MSE(θ^)=Var[θ^]. An unbiased estimator such that MSE(θ^) is a minimum value among all unbiased estimators for θ is called a minimum variance unbiased estimator, abbreviated MVUE, or uniformly minimum variance unbiased estimator, abbreviated UMVU estimator.

Example. Suppose X1,X2,,Xn are iid random variablesMathworldPlanetmath (n independentPlanetmathPlanetmath measurements of the radius of a coin, etc…) from a normal distributionMathworldPlanetmath N(μ,σ2) (for example, μ would be the true radius of the coin, and σ2 would be the error component of the measurements). Suppose X¯ (=X¯n) is the sample meanMathworldPlanetmath. Then X¯ is an unbiased estimator, so that


Remark. The square root of MSE is called the “root mean square error”, or rms error for short.

Title mean square error
Canonical name MeanSquareError
Date of creation 2013-03-22 12:07:42
Last modified on 2013-03-22 12:07:42
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 62J10
Classification msc 94A12
Synonym MSE
Synonym MVUE
Synonym UMVU
Synonym UMVUE
Synonym uniformly minimum variance unbiased
Related topic MeanSquareDeviation
Defines minimum variance unbiased estimator
Defines rms error
Defines root-mean-square
Defines root mean square
Defines rms