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In statistics the **mean squared prediction error** of a smoothing procedure is the expected sum of squared deviations of the fitted values from the (unobservable) function . If the smoothing procedure has operator matrix , then

The MSPE can be decomposed into two terms just like mean squared error is decomposed into bias and variance; however for MSPE one term is the sum of squared biases of the fitted values and another the sum of variances of the fitted values:

Note that knowledge of is required in order to calculate MSPE exactly.

## Estimation of MSPE[]

For the model where , one may write

The first term is equivalent to

Thus,

If is known or well-estimated by , it becomes possible to estimate MSPE by

Colin Mallows advocated this method in the construction of his model selection statistic Cp, which is a normalized version of the estimated MSPE:

where comes from that fact that the number of parameters estimated for a parametric smoother is given by , and is in honor of Cuthbert Daniel.

## See also[]

- Mean squared error
- Errors and residuals in statistics
- Law of total variance

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