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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

##### MSC:
 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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