Mean squared prediction error

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:


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

Estimation of MSPE

For the model yi = g(xi) + σεi where \varepsilon_i\sim\mathcal{N}(0,1), one may write


The first term is equivalent to




If σ2 is known or well-estimated by \widehat{\sigma}^2, 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 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.

See also

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