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This chapter considers the choice of explanatory variables to include in the linearPredictor predictorLinear predictor  \(x^T\beta \) . We start with models for all of which p, the dimension of \(\beta \) , is \( < n.\) The problem arises specifically when some variables are nearly collinear when the significance of a variable in the model may depend strongly on what other variables are included. Section 9.3.1 derives Mallow’s \(C_p\) from Akaike’s AIC; models with more parameters are penalized. RobustnessRobustness, together with the detection of influential observations, is provided by the generalized candlestick plot, illustrated by three data analyses. For the rest of the chapter we take \(n < p.\) Section 9.4.1 describes two regularizationsRegularization: the LASSO estimates \(\beta \) as the minimizer of a linear combination of the \(L_2\) norm of the residualsResiduals and the \(L_1\) norm of the parameter estimatesParameter estimates. The method provides model selection, the number of parameter estimatesParameter estimates set to zero depending on a parameter \(\lambda \) . In ridge regressionRidge regression both terms use the \(L_2\) norm, the parameter controlling the shrinkage of the parameter estimatesParameter estimates. Neither method is robust. Section 9.4.2 describes sparse LTSLeast Trimmed Squares (LTS) which adds an \(L_1\) penalty term with penalty parameter \(\lambda \) to LTSLeast Trimmed Squares (LTS) estimation. In Sect. 9.4.3 the parameter \(\lambda \) is estimated, for the cancer data of Sect. 9.4.1, by monitoring. Seven explanatory variables occur in many of the selected models. These are subjected to robust model selection in Sect.  10.6 .

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Model Selection

  • 章节

  • 出版模式：

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  • 发表日期：

    2025年

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  • Anthony C. Atkinson,
  • Marco Riani,
  • Aldo Corbellini,
  • Domenico Perrotta,
  • Valentin Todorov

摘要

This chapter considers the choice of explanatory variables to include in the linearPredictor predictorLinear predictor  \(x^T\beta \) . We start with models for all of which p, the dimension of \(\beta \) , is \( < n.\) The problem arises specifically when some variables are nearly collinear when the significance of a variable in the model may depend strongly on what other variables are included. Section 9.3.1 derives Mallow’s \(C_p\) from Akaike’s AIC; models with more parameters are penalized. RobustnessRobustness, together with the detection of influential observations, is provided by the generalized candlestick plot, illustrated by three data analyses. For the rest of the chapter we take \(n < p.\) Section 9.4.1 describes two regularizationsRegularization: the LASSO estimates \(\beta \) as the minimizer of a linear combination of the \(L_2\) norm of the residualsResiduals and the \(L_1\) norm of the parameter estimatesParameter estimates. The method provides model selection, the number of parameter estimatesParameter estimates set to zero depending on a parameter \(\lambda \) . In ridge regressionRidge regression both terms use the \(L_2\) norm, the parameter controlling the shrinkage of the parameter estimatesParameter estimates. Neither method is robust. Section 9.4.2 describes sparse LTSLeast Trimmed Squares (LTS) which adds an \(L_1\) penalty term with penalty parameter \(\lambda \) to LTSLeast Trimmed Squares (LTS) estimation. In Sect. 9.4.3 the parameter \(\lambda \) is estimated, for the cancer data of Sect. 9.4.1, by monitoring. Seven explanatory variables occur in many of the selected models. These are subjected to robust model selection in Sect.  10.6 .