Optimal cross-validation in density estimation with the L2L^2-loss

We analyze the performance of cross-validation (CV) in the density estimation framework with two purposes: (i) risk estimation and (ii) model selection. The main focus is given to the so-called leave-$p$-out CV procedure (Lpo), where $p$ denotes the cardinality of the test set. Closed-form expressions are settled for the Lpo estimator of the risk of projection estimators. These expressions provide a great improvement upon $V$-fold cross-validation in terms of variability and computational complexity. From a theoretical point of view, closed-form expressions also enable to study the Lpo performance in terms of risk estimation. The optimality of leave-one-out (Loo), that is Lpo with $p=1$, is proved among CV procedures used for risk estimation. Two model selection frameworks are also considered: estimation, as opposed to identification. For estimation with finite sample size $n$, optimality is achieved for $p$ large enough [with $p/n=o(1)$] to balance the overfitting resulting from the structure of the model collection. For identification, model selection consistency is settled for Lpo as long as $p/n$ is conveniently related to the rate of convergence of the best estimator in the collection: (i) $p/n\to1$ as $n\to+\infty$ with a parametric rate, and (ii) $p/n=o(1)$ with some nonparametric estimators. These theoretical results are validated by simulation experiments.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1240 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

An upper bound on the number of rational points of arbitrary projective varieties over finite fields

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non equidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety

Three new species of Eupetersia Blüthgen, 1928 (Hymenoptera, Halictidae) from the Oriental Region

Three new species, Eupetersia (Nesoeupetersia) singaporensis sp. nov., collected in a mangrove swamp in Singapore, and Eupetersia (Nesoeupetersia) sabahensis sp. nov., collected in the mountains of Sabah, Borneo, and Eupetersia (Nesoeupetersia) yanegai sp. nov., collected in Thailand, are described. This genus is more diversified in the sub-Saharan region, including Madagascar. The only other Oriental species, E. nathani (Baker, 1974), was described from India and is diagnosed and re-illustrated here