Atkinson,A.C., Riani,M., Corbellini,A., Perrotta D., and Todorov,V. (2024), Applied Robust Statistics through the Monitoring Approach, Heidelberg: Springer Nature.

This chapter considers the choice of explanatory variables to include in the linear predictor $x^T\beta$. We start with models for all of which $p$, the dimension of $\beta$, is $p < n$. 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. Robustness 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 regularizations: the LASSO estimates $\beta$ as the minimizer of a linear combination of the $L_2$ norm of the residuals and the $L_1$ norm of the parameter estimates. The method provides model selection, the number of parameter estimates set to zero depending on a parameter $\lambda$. In ridge regression both terms use the $L_2$ norm, the parameter $\lambda$ controlling the shrinkage of the parameter estimates. Neither method is robust. Section 9.4.2 describes sparse LTS which adds an $L_1$ penalty term with penalty parameter $\lambda$ to LTS estimation. In 9.4.3 the parameter $\lambda$ is estimated, for the cancer data of 9.4.1, by monitoring. Seven explanatory variables occur in many of the selected models. These are subjected to robust model selection in section 10.6.

FileName	Description	Open in MATLAB on line	Jupiter notebook
Cement.m	Cement data. This file creates Tables 9.1 and 9.2.		[ipynb]
NCI60sparseLTS.R	NIC60 data: Least Absolute Shrinkage and Selection Operator (LASSO) This file creates Figures 9.3, 9.4, 9.5, 9.6 and 9.7.	Open
Ozone.m	Ozone data (reduced and full). This file creates Figures 9.1 and 9.2		[ipynb]