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Consistency of cross validation for comparing regression procedures. (English) Zbl 1129.62039

Summary: Theoretical developments on cross validation (CV) have mainly focused on selecting one among a list of finite-dimensional models (e.g., subset or order selection in linear regression) or selecting a smoothing parameter (e.g., bandwidth for kernel smoothing). However, little is known about consistency of cross validation when applied to compare between parametric and nonparametric methods or within nonparametric methods. We show that under some conditions, with an appropriate choice of data splitting ratio, cross validation is consistent in the sense of selecting the better procedure with probability approaching 1.
Our results reveal interesting behavior of cross validation. When comparing two models (procedures) converging at the same nonparametric rate, in contrast to the parametric case, it turns out that the proportion of data used for evaluation in CV does not need to be dominating in size. Furthermore, it can even be of a smaller order than the proportion for estimation while not affecting the consistency property.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
62G05 Nonparametric estimation

Keywords:

model selection

Software:

alr3
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References:

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