NationMaster - Encyclopedia: Mean squared error

In statistics the mean squared error of an estimator T of an unobservable parameter θ is A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In statistics, an estimator is a function of the known data that is used to estimate an unknown parameter; an estimate is the result from the actual application of the function to a particular set of data. ...

$operatorname{MSE}(T)=operatorname{E}((T-theta)^2),$

i.e., it is the expected value of the square of the "error". The "error" is the amount by which the estimator differs from the quantity to be estimated. The mean squared error satisfies the identity In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical...

$operatorname{MSE}(T)=operatorname{var}(T)+(operatorname{bias}(T))^2$

where

$operatorname{bias}(T)=operatorname{E}(T)-theta,$

i.e., the bias is the amount by which the expected value of the estimator differs from the unobservable quantity to be estimated. In statistics, the term bias is used for two different concepts. ...

Here is a concrete example. Suppose

$X_1,dots,X_nsimoperatorname{N}(mu,sigma^2),$

i.e., this is a random sample of size n from a normally distributed population. Two estimators of σ² are sometimes used (as are others): The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ...

where

$overline{X}=(X_1+cdots+X_n)/n$

is the "sample mean". The first of these estimators is the maximum likelihood estimator, and is biased, i.e., its bias is not zero, but has a smaller variance than the second, which is unbiased. The smaller variance compensates somewhat for the bias, so that the mean squared error of the biased estimator is slightly smaller than that of the unbiased estimator. Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. ...

The root mean squared error (RMSE) is simply the square root of the MSE.

External Links

Program for MSE measurements in images and video

Category: Estimation theory

Results from FactBites:

Mean squared error - Wikipedia, the free encyclopedia (203 words)
	In statistics the mean squared error of an estimator T of an unobservable parameter θ is
	The "error" is the amount by which the estimator differs from the quantity to be estimated.
	The smaller variance compensates somewhat for the bias, so that the mean squared error of the biased estimator is slightly smaller than that of the unbiased estimator.

3.2 Error Measurements (691 words)
	This error output, taken from the output from the Naive Bayes learner model, is typical of error output for models that have numerical output attributes.
	Mean absolue error is the average of the difference between predicted and actual value in all test cases; it is the average prediction error.
	Relative squared error is the total squared error made relative to what the error would have been if the prediction had been the average of the absolute value.

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