  This work develops central limit theorems for cross-validation and consistent
estimators of its asymptotic variance under weak stability conditions on the
learning algorithm. Together, these results provide practical,
asymptotically-exact confidence intervals for $k$-fold test error and valid,
powerful hypothesis tests of whether one learning algorithm has smaller
$k$-fold test error than another. These results are also the first of their
kind for the popular choice of leave-one-out cross-validation. In our real-data
experiments with diverse learning algorithms, the resulting intervals and tests
outperform the most popular alternative methods from the literature.
