We consider choosing an estimator or model from a given class by cross validation consisting of holding a nonneglible fraction of the observations out as a test set. We derive bounds that show that... Show moreWe consider choosing an estimator or model from a given class by cross validation consisting of holding a nonneglible fraction of the observations out as a test set. We derive bounds that show that the risk of the resulting procedure is (up to a constant) smaller than the risk of an oracle plus an error which typically grows logarithmically with the number of estimators in the class. We extend the results to penalized cross validation in order to control unbounded loss functions. Applications include regression with squared and absolute deviation loss and classification under Tsybakov’s condition. Show less
We describe explicit methods of exhibiting elements of the Brauer groups of diagonal quartic surfaces. Using these methods, we compute the algebraic Brauer–Manin obstruction in two contrasting... Show moreWe describe explicit methods of exhibiting elements of the Brauer groups of diagonal quartic surfaces. Using these methods, we compute the algebraic Brauer–Manin obstruction in two contrasting examples. In the second example, the obstruction is found to be trivial but a computer search reveals no points of small height on the surface. Show less
