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High-dimensional variable selection. (English) Zbl 1173.62054
Summary: This paper explores the following question: what kind of statistical guarantees can be given when doing variable selection in high-dimensional models? In particular, we look at the error rates and power of some multi-stage regression methods. In the first stage we fit a set of candidate models. In the second stage we select one model by cross-validation. In the third stage we use hypothesis testing to eliminate some variables. We refer to the first two stages as “screening” and the last stage as “cleaning.” We consider three screening methods: the lasso, marginal regression, and forward stepwise regression. Our method gives consistent variable selection under certain conditions.
MSC:
62J05Linear regression
62F12Asymptotic properties of parametric estimators
62H12Multivariate estimation
62F10Point estimation
62J07Ridge regression; shrinkage estimators
62P10Applications of statistics to biology and medical sciences