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


 * $$\operatorname{MSPE}(L)=\operatorname{E}\left[\sum_{i=1}^n\left( g(x_i)-\widehat{g}(x_i)\right)^2\right].$$

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:


 * $$\operatorname{MSPE}(L)=\sum_{i=1}^n\left(\operatorname{E}\left[\widehat{g}(x_i)\right]-g(x_i)\right)^2+\sum_{i=1}^n\operatorname{var}\left[\widehat{g}(x_i)\right].$$

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

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


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

The first term is equivalent to


 * $$\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,


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

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


 * $$\operatorname{\widehat{MSPE}}(L)=\sum_{i=1}^n\left(y_i-\widehat{g}(x_i)\right)^2-\widehat{\sigma}^2\left(n-2\operatorname{tr}\left[L\right]\right).$$

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


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

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