# Mean Squared Error (MSE) Measure

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A Mean Squared Error (MSE) Measure is a point estimator evaluation metric that is based on the average of the estimator's squared errors.

## References

### 2015

1. Lehmann, E. L.; Casella, George (1998). Theory of Point Estimation (2nd ed.). New York: Springer. ISBN 978-0-387-98502-2. MR 1639875

### 2013

• (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Mean_square_error#Definition_and_basic_properties
• If $\hat{Y}$ is a vector of n predictions, and $Y$ is the vector of the true values, then the MSE of the predictor is: :$MSE=\frac{1}{n}\sum_{i=1}^n(\hat{Y_i} - Y_i)^2.$ This is a known, computed quantity given a particular sample (and hence is sample-dependent).

The MSE of an estimator $\hat{\theta}$ with respect to the unknown parameter $\theta$ is defined as  :$\operatorname{MSE}(\hat{\theta})=\operatorname{E}\big[(\hat{\theta}-\theta)^2\big].$ This definition depends on the unknown parameter, and the MSE in this sense is a property of an estimator (of a method of obtaining an estimate).

The MSE is equal to the sum of the variance and the squared bias of the estimator or of the predictions. In the case of the MSE of an estimator,[1] :$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2.$ The MSE thus assesses the quality of an estimator or set of predictions in terms of its variation and degree of bias.

Since MSE is an expectation, it is not a random variable. It may be a function of the unknown parameter $\theta$, but it does not depend on any random quantities. However, when MSE is computed for a particular estimator of $\theta$ the true value of which is not known, it will be subject to estimation error. In a Bayesian sense, this means that there are cases in which it may be treated as a random variable.

1. Wackerly, Dennis; Scheaffer, William (2008). Mathematical Statistics with Applications (7 ed.). Belmont, CA, USA: Thomson Higher Education. ISBN 0-495-38508-5.

### 2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Mean_squared_error
• QUOTE: In Statistics, the mean squared error or MSE of an Estimator is one of many ways to quantify the amount by which an Estimator differs from the true value of the quantity being estimated. As a loss function, MSE is called squared error loss. MSE measures the average of the square of the "error." The error is the amount by which the estimator differs from the quantity to be estimated. The difference occurs because of randomness or because the estimator doesn't account for information that could produce a more accurate estimate.[1]
The MSE is the second moment (about the origin) of the error, and thus incorporates both the variance of the estimator and its bias. For an Unbiased Estimator, the MSE is the variance. Like the variance, MSE has the same unit of measurement as the square of the quantity being estimated. In an analogy to Standard Deviation, taking the square root of MSE yields the Root Mean Squared Error or RMSE, which has the same units as the quantity being estimated; for an unbiased estimator, the RMSE is the square root of the variance, known as the standard error.
1. George Casella & E.L. Lehmann, "Theory of Point Estimation". Springer, (1999)

### 1981

• (Sheiner & Beal, 1981) ⇒ Lewis B. Sheiner, and Stuart L. Beal. (1981). “Some Suggestions for Measuring Predictive Performance.” In: Journal of Pharmacokinetics and Pharmacodynamics, 9(4). doi:10.1007/BF01060893.
• ABSTRACT: The performance of a prediction or measurement method is often evaluated by computing the correlation coefficient and/or the regression of predictions on true (reference) values. These provide, however, only a poor description of predictive performance. The mean squared prediction error (precision) and the mean prediction error (bias) provide better descriptions of predictive performance. These quantities are easily computed, and can be used to compare prediction methods to absolute standards or to one another. The measures, however, are unreliable when the reference method is imprecise. The use of these measures is discussed and illustrated.