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Loss function

In the theory of point estimation, a loss function quantifies the losses associated to the errors committed while estimating a parameter. Often the expected value of the loss, called statistical risk, is used to compare two or more estimators: in such comparisons, the estimator having the least expected loss is usually deemed preferable.

Definition

The following is a possible definition.

Definition Let $\theta _{0}$ be an unknown parameter and $\widehat{\theta }$ an estimate of $\theta _{0}$. The estimation error is the difference[eq1]The loss function is a function mapping estimation errors to the set of real numbers.

Examples

Let [eq2] denote the Euclidean norm. Commonly used loss functions are:

In both cases, the larger the estimation error is, the larger is the loss. The expected value of the former is called mean absolute error (MAE), while the expectation of the latter is known as mean squared error (MSE).

More details

Loss functions, estimation errors and statistical risk are explained in more detailed in the lecture entitled Point estimation.

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