Mean squared prediction error: Difference between revisions

Revision as of 22:52, 7 January 2013

Revision as of 22:44, 30 March 2013

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{{Unreferenced|date=December 2009}}

In [[statistics]] the '''mean squared prediction error''' of a [[smoothing]] or [[curve fitting]] procedure is the expected sum of squared deviations of the fitted values <math>\widehat{g}</math> from the (unobservable) function <math>''g</math>''. If the smoothing procedure has [[operator matrix]] <math>''L</math>'', then

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

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:<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 <math>''g</math>'' is required in order to calculate MSPE exactly.

==Estimation of MSPE==

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

==See also==