Mean squared prediction error

In statistics the mean squared prediction error of a smoothing procedure is the expected sum of squared deviations of the fitted values ${\hat {g}}$ from the (unobservable) function $g$. If the smoothing procedure has operator matrix $L$, then

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

The MSPE can be decomposed into two terms:

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

This decomposition is analogous to that of mean squared error 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:

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

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

Estimation of MSPE

For the model $y_{i}=g(x_{i})+\sigma \epsilon _{i}$ where $\epsilon _{i}\sim {\mathcal {N}}(0,1)$, the first term can be rewritten as

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

whereas the second term is $\sigma ^{2}\operatorname {tr} \left[L'L\right]$. Putting it all together,

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

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

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

Colin Mallows advocated this method in the construction of his model selection statistic $C_{p}$, which is a normalized version of the estimated MSPE:

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

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