## Choice of V for V-Fold Cross-Validation in Least-Squares Density Estimation

Sylvain Arlot, Matthieu Lerasle; 17(208):1−50, 2016.

### Abstract

This paper studies $V$-fold cross-validation for model selection in least-squares density estimation. The goal is to provide theoretical grounds for choosing $V$ in order to minimize the least-squares loss of the selected estimator. We first prove a non-asymptotic oracle inequality for $V$-fold cross-validation and its bias-corrected version ($V$-fold penalization). In particular, this result implies that $V$-fold penalization is asymptotically optimal in the nonparametric case. Then, we compute the variance of $V$-fold cross-validation and related criteria, as well as the variance of key quantities for model selection performance. We show that these variances depend on $V$ like $1+4/(V-1)$, at least in some particular cases, suggesting that the performance increases much from $V=2$ to $V=5$ or $10$, and then is almost constant. Overall, this can explain the common advice to take $V=5\,$---at least in our setting and when the computational power is limited---, as supported by some simulation experiments. An oracle inequality and exact formulas for the variance are also proved for Monte- Carlo cross-validation, also known as repeated cross-validation, where the parameter $V$ is replaced by the number $B$ of random splits of the data.

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