The Annals of Statistics

The Risk Inflation Criterion for Multiple Regression

Dean P. Foster and Edward I. George

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A new criterion is proposed for the evaluation of variable selection procedures in multiple regression. This criterion, which we call the risk inflation, is based on an adjustment to the risk. Essentially, the risk inflation is the maximum increase in risk due to selecting rather than knowing the "correct" predictors. A new variable selection procedure is obtained which, in the case of orthogonal predictors, substantially improves on AIC, $C_p$ and BIC and is close to optimal. In contrast to AIC, $C_p$ and BIC which use dimensionality penalties of 2, 2 and $\log n$, respectively, this new procedure uses a penalty $2 \log p$, where $p$ is the number of available predictors. For the case of nonorthogonal predictors, bounds for the optimal penalty are obtained.

Article information

Ann. Statist. Volume 22, Number 4 (1994), 1947-1975.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62C99: None of the above, but in this section
Secondary: 62J05: Linear regression 62C20: Minimax procedures

Decision theory minimax model selection multiple regression risk variable selection


Foster, Dean P.; George, Edward I. The Risk Inflation Criterion for Multiple Regression. Ann. Statist. 22 (1994), no. 4, 1947--1975. doi:10.1214/aos/1176325766.

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