Mathematics – Statistics Theory
Scientific paper
2008-03-20
Annals of Statistics 2007, Vol. 35, No. 6, 2450-2473
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/009053607000000514 the Annals of Statistics (http://www.imstat.org/aos/) by the Inst
Scientific paper
10.1214/009053607000000514
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.
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