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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 ${\displaystyle {\widehat {g}}}$ and the (unobservable) function g. If the smoothing procedure has operator matrix L, then

${\displaystyle \operatorname {MSPE} (L)=\operatorname {E} \left[\left(g(x_{i})-{\widehat {g}}(x_{i})\right)^{2}\right].}$

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:

${\displaystyle \operatorname {MSPE} (L)=\sum _{i=1}^{n}\left(\operatorname {E} \left[{\widehat {g}}(x_{i})\right]-g(x_{i})\right)^{2}+\sum _{i=1}^{n}\operatorname {var} \left[{\widehat {g}}(x_{i})\right].}$

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

For the model ${\displaystyle y_{i}=g(x_{i})+\sigma \varepsilon _{i}}$ where ${\displaystyle \varepsilon _{i}\sim {\mathcal {N}}(0,1)}$, one may write

${\displaystyle \operatorname {MSPE} (L)=g'(I-L)'(I-L)g+\sigma ^{2}\operatorname {tr} \left[L'L\right].}$

The first term is equivalent to

${\displaystyle \sum _{i=1}^{n}\left(\operatorname {E} \left[{\widehat {g}}(x_{i})\right]-g(x_{i})\right)^{2}=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\operatorname {tr} \left[\left(I-L\right)'\left(I-L\right)\right].}$

Thus,

${\displaystyle \operatorname {MSPE} (L)=\operatorname {E} \left[\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}\right]-\sigma ^{2}\left(n-2\operatorname {tr} \left[L\right]\right).}$

If ${\displaystyle \sigma ^{2}}$ is known or well-estimated by ${\displaystyle {\widehat {\sigma }}^{2}}$, it becomes possible to estimate MSPE by

${\displaystyle \operatorname {\widehat {MSPE}} (L)=\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}-{\widehat {\sigma }}^{2}\left(n-2\operatorname {tr} \left[L\right]\right).}$

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

${\displaystyle C_{p}={\frac {\sum _{i=1}^{n}\left(y_{i}-{\widehat {g}}(x_{i})\right)^{2}}{{\widehat {\sigma }}^{2}}}-n+2\operatorname {tr} \left[L\right].}$

where p comes from that fact that the number of parameters p estimated for a parametric smoother is given by ${\displaystyle p=\operatorname {tr} \left[L\right]}$, and C is in honor of Cuthbert Daniel.^{[citation needed]}

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