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We determine the asymptotically optimal choice of the parameter ν for classifiers of ν-support vector machine (ν-SVM) type which has been introduced by Scholkopf et al. (2000). It turns out that ν should be a close upper estimate of twice the optimal Bayes risk provided that the classifier uses a so-called universal kernel such as the Gaussian RBF kernel. Moreover, several experiments show that this result can be used to implement some modified cross validation procedures which improve standard cross validation for ν-SVMs.