Some comments on C_p

By C. L. Mallows. 1973

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Hyperparameter optimization with approximate gradient (2016, background): “This criterion can be a goodness of ﬁt on unseen data, such as a cross-validation loss, or some criteria of model quality on the train set such as SURE (Stein, 1981), AIC/BIC (Liu and Yang, 2011) or Mallows C p (Mallows, 1973), to name a few.”
The Variable Selection Problem (2000, methodology): “Motivated as an unbiased estimate of predictive accuracy of the 7th model, Mallows (1973) recommended using C, plots to help gauge subset selection (see also Mallows 1995).”
Effective degrees of freedom: a flawed metaphor (2013, unknown): “Both plots show that the DF for a number of subset sizes is greater than 15, and we know from standard OLS theory that the DF for subset size 15 is exactly 15 in both cases.”
Parametric or nonparametric? A parametricness index for model selection (2011, background): “It is well known that AIC [2], C p [47] and FPE [1, 58] have an asymptotic optimality property which says the accuracy of the estimator based on the selected model is asymptotically the same as the best candidate model when the true model is inﬁnite dimensional.”
Bivariate Binomial Spatial Modeling of Loa loa Prevalence in Tropical Africa (2008, methodology): “These include cross-validation (CV) or generalized CV (GCV) (Craven and Wahba 1979), Cp (Mallows 1973), Akaike information (Akaike 1973) and restricted or unrestricted maximum likelihood (REML) (Ruppert, Wand, and Carroll 2003; Kammann and Wand 2003) criteria.”
Risk bounds for the non-parametric estimation of Lévy processes (2006, background): “…in (2.14), the bias term, accounts for the distance of the unknown function s to the model S, while the second term, the variance term, measures the error of estimation within the linear model S. Notice that (2.13) is finite because s is assumed bounded on D and thus, (2.15) E [ χ2 ] ≤ ‖s‖∞ d T .”
A Note on the Selection of Time Series Models (2005, unknown): “The C p criterion of Mallows (1973) replaces σ 2 by a consistent estimate (say, σ 2 ) that is the same across models to be compared giving C p = τ σ 2 k σ 2 + (2k − N ).”
Variable Selection in Predictive Regressions (2013, background): “The CP criterion defines XA as the subset of explanatory variables that corresponds to the lowest point in the plot of CP against p. Mallows (1973) does not recommend to blindly follow this practice because the rule will not be reliable when a large number of subsets are close competitors to the…”
Akaike's Information Criterion, Cp and Estimators of Loss for Elliptically Symmetric Distributions (2014, methodology): “This practice changed with Mallows’s automated criterion known as Cp (Mallows, 1973).”
Panel Data Models with Grouped Factor Structure Under Unknown Group Membership (2013, methodology): “We can regard the proposed criterion as a generalization of the Cp criterion of Mallows (1973) in the panel data context with unobservable interactive effects.”
Structured Prediction Cascades (2010, methodology): “Standard statistical model selection techniques (Mallows, 1973; Vapnik and Chervonenkis, 1974; Akaike, 1974; Devroye et al., 1996; Barron et al., 1999; Bartlett et al., 2002) explore a hierarchy of models of increasing complexity primarily to minimize expected error, without much concern for the…”