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单词 MeanSquareError
释义

mean square error


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

MSE(θ^):=E[(θ^-θ)2].

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

MSE(θ^)=Var[θ^]+Bias(θ^)2.

If θ^ is an unbiased estimatorMathworldPlanetmath, then its mean squareerror 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 iscalled a minimum variance unbiased estimator, abbreviated MVUE, or uniformly minimum variance unbiased estimator, abbreviated UMVU estimator.

Example. Suppose X1,X2,,Xn are iid randomvariablesMathworldPlanetmath (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 bethe error component of the measurements). Suppose X¯(=X¯n) is the sample meanMathworldPlanetmath. Then X¯ is anunbiased estimator, so that

MSE(X¯)=Var[X¯]=Var[1ni=1nXi]=1n2(i=1nσ2)=σ2n.

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

Titlemean square error
Canonical nameMeanSquareError
Date of creation2013-03-22 12:07:42
Last modified on2013-03-22 12:07:42
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id11
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 62J10
Classificationmsc 94A12
SynonymMSE
SynonymMVUE
SynonymUMVU
SynonymUMVUE
Synonymuniformly minimum variance unbiased
Related topicMeanSquareDeviation
Definesminimum variance unbiased estimator
Definesrms error
Definesroot-mean-square
Definesroot mean square
Definesrms
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