Mean squared error

The mean square error or mean square error (german mean squared error and therefore with " MSE " for short ) is a term of mathematical statistics. ( Or more generally of functionals of them) with the mean square deviation, the deviation of an estimator of the value to be estimated can be calculated.

Definition

Let a random variable and a measurable function of these variables. If so, the true parameters to be estimated, then the root mean square deviation of the estimator for being defined as follows:

In the classical case of real-valued functions that is then:

Here, the distortion vanishes in the case of unbiasedness where the MSE value and the variance are the same.

Interpretation

A small root mean square deviation means in the classical case, that the same bias and variance of the estimator are small. It is located in the middle of the estimates so near the to be estimated functional ( low bias) and white at the same time that the estimates little scatter ( low variance ) and most likely also lie close to their expected value.

The MSE, it is possible to compare with each other estimation methods. The idea is that it may be advantageous to favor a slightly distorted estimator that it has a much smaller variance. In this case, the estimation method having the smaller MSE is generally regarded as the better.

The problem is that the MSE to be estimated from the unknown population parameter dependent.

Example

A typical case is the estimate of the mean of a normal distribution. We assume that random variables exist that are each normally distributed with unknown mean and variance 1. The classical estimator is the sample mean. Here the distortion is zero:

Because the empirical mean is unbiased for. Since even a normal distribution with mean and variance, it follows

Extension to arbitrary loss functions

A generalization of the mean square error is obtained, in the definition in place of the square distance of unknown functional estimator and by substituting any other function,

  • Symmetrical,
  • Has values ​​in and
  • Is convex in both components.

Pictures of this style are called loss function, the risk of an estimator is then defined as

Functional (mathematics)
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