Mean Squared Error

In Statistical Analysis, the Mean Squared Error (MSE), also called Mean Squared Deviation (MSD), of an estimator measures the average of the square of the errors. This is the average squared difference between the estimated values and the true value.

MSE is a risk function, corresponding to the expected value of the Squared Error Loss. It’s a measure of quality of the estimator.

It is always a positive value that decreases as the error approaches 0.

The MSE is the second moment (about the origin) of the error. This means it incorporates the Variance of the estimator (how the estimates are spread from one data sample) and it’s bias (how far off the average estimated value is from the true value).

For an unbiased estimator, the MSE is the variance of the estimator.

Definition and Properties

The MSE either asses the quality of a predictor or of an estimator. The definition of an MSE differs according to whether one is describing a predictor or an estimator.

Predictor

If a vector of $n$ is generated from a sample data of $n$ data points on all variables, and $Y$ is the vector of observed values being predicted, and $\hat{Y}$ being the predicted values, then the MSE is computed as: $MSE=\frac{1}{n}{\sum }_{i=1}^n(Y_i-\widehat{Y_i})$

Estimator

The MSE of an estimator with respect to an unknown parameter $\theta$ is defined as the expected value of $\theta$ of the squared difference between the estimator and the parameter. $MSE(\hat{\theta })=E_{\theta }[(\hat{\theta }-\theta {)}^2]$ The MSE can be written as the sum of the variance of the estimator and the squared bias of the estimator. This means that, for an unbiased estimator, the MSE and the variance are equivalent.