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Asymptotic optimality for \(C_ p\), \(C_ L\), cross-validation and generalized cross-validation: Discrete index set. (English) Zbl 0653.62037
Let \(X_ i\) be independent random variables with means \(m_ i\), \(1\leq i\leq n\), and common variance \(\sigma^ 2\). The mean vector m is to be estimated by an element of a given finite set of linear estimators \(\hat m,\) \(\hat m\) shall be chosen such that \(L_ n(\hat m)=n^{-1} \| m- \hat m\|^ 2\) is a minimum.
The author studies the asymptotic behaviour of three procedures for selecting \(\hat m:\) Mallows’ \(C_ L\), a general cross validation and the delete-one cross validation. For two classes of examples - model selection and nearest neighbour nonparametric regression - both covered by this setting, sufficient conditions are given for \(L_ n(\hat m)[\min_{\hat m}L_ n(\hat m)]^{-1}\) to converge in probability to one. The connections between the three cross validation procedures are discussed in detail.
Reviewer: O.Krafft

62G99 Nonparametric inference
62J99 Linear inference, regression
62G05 Nonparametric estimation
62J05 Linear regression; mixed models
62J07 Ridge regression; shrinkage estimators (Lasso)
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