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{{Unreferenced|date=December 2009}}
{{expert-subject|statistics|reason=no source, and notation/definition problems regarding ''L''}}
In [[statistics]] the '''mean squared prediction error''' of a [[smoothing]] or [[curve fitting]] procedure is the expected value of the squared difference between the fitted values <math>\widehat{g}</math> and the (unobservable) function ''g''. If the smoothing procedure has [[operator matrix]] ''L'', then

:<math>\operatorname{MSPE}(L)=\operatorname{E}\left[\left( g(x_i)-\widehat{g}(x_i)\right)^2\right].</math>

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:

:<math>\operatorname{MSPE}(L)=\sum_{i=1}^n\left(\operatorname{E}\left[\widehat{g}(x_i)\right]-g(x_i)\right)^2+\sum_{i=1}^n\operatorname{var}\left[\widehat{g}(x_i)\right].</math>

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

==Estimation of MSPE==
For the model <math>y_i=g(x_i)+\sigma\varepsilon_i</math> where <math>\varepsilon_i\sim\mathcal{N}(0,1)</math>, one may write

:<math>\operatorname{MSPE}(L)=g'(I-L)'(I-L)g+\sigma^2\operatorname{tr}\left[L'L\right].</math>

The first term is equivalent to 

:<math>\sum_{i=1}^n\left(\operatorname{E}\left[\widehat{g}(x_i)\right]-g(x_i)\right)^2
=\operatorname{E}\left[\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2\right]-\sigma^2\operatorname{tr}\left[\left(I-L\right)'\left(I-L\right)\right].</math> 

Thus,

:<math>\operatorname{MSPE}(L)=\operatorname{E}\left[\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2\right]-\sigma^2\left(n-2\operatorname{tr}\left[L\right]\right).</math>

If <math>\sigma^2</math> is known or well-estimated by <math>\widehat{\sigma}^2</math>, it becomes possible to estimate MSPE by

:<math>\operatorname{\widehat{MSPE}}(L)=\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2-\widehat{\sigma}^2\left(n-2\operatorname{tr}\left[L\right]\right).</math>

[[Colin Mallows]] advocated this method in the construction of his model selection statistic [[Mallows's Cp|''C<sub>p</sub>'']], which is a normalized version of the estimated MSPE:

:<math>C_p=\frac{\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2}{\widehat{\sigma}^2}-n+2\operatorname{tr}\left[L\right].</math>

where ''p'' comes from that fact that the number of parameters ''p'' estimated for a parametric smoother is given by <math>p=\operatorname{tr}\left[L\right]</math>, and ''C'' is in honor of [[Cuthbert Daniel]].{{citation needed|date=March 2013}}

==See also==
* [[Mean squared error]]
* [[Errors and residuals in statistics]]
* [[Law of total variance]]

{{DEFAULTSORT:Mean Squared Prediction Error}}
[[Category:Point estimation performance]]
[[Category:Statistical deviation and dispersion]]
[[Category:Loss functions]]