In model selection, several types of cross-validation are commonly used and many variants have been introduced. While consistency of some of these methods has been proven, their rate of convergence to the oracle is generally still unknown. Until now, an asymptotic analysis of crossvalidation able to answer this question has been lacking. Existing results focus on the ''pointwise'' estimation of the risk of a single estimator, whereas analysing model selection requires understanding how the CV risk varies with the model. In this article, we investigate the asymptotics of the CV risk in the neighbourhood of the optimal model, for trigonometric series estimators in density estimation. Asymptotically, simple validation and ''incomplete'' V --fold CV behave like the sum of a convex function fn and a symmetrized Brownian changed in time W gn/V. We argue that this is the right asymptotic framework for studying model selection.

翻译：在选择模型时,通常使用几种交叉验证方法,并采用了许多变量。虽然已经证明了其中一些方法的一致性,但它们与神器的趋同率一般仍不得而知。直到现在,还没有对能够回答这一问题的交叉验证进行无症状分析。现有结果侧重于“点针”估计单一估计器的风险,而分析模型选择则需要了解CV风险与模型的不同。在本条中,我们调查了在最佳模型的周边,即密度估计的三维测量序列估计器中,CV风险的无症状。简单、简单验证和“不完全”V——CV行为类似于在Wgn/V时变化的convex函数总和和和相匹配的Brownian。我们说,这是研究模型选择的无症状框架。