Parzen windows for multi-class classification

We consider the multi-class classification problem in learning theory. A learning algorithm by means of Parzen windows is introduced. Under some regularity conditions on the conditional probability for each class and some decay condition of the marginal distribution near the boundary of the input space, we derive learning rates in terms of the sample size, window width and the decay of the basic window. The choice of the window width follows from bounds for the sample error and approximation error. A novelly defined splitting function for the multi-class classification and a comparison theorem, bounding the excess misclassification error by the norm of the difference of function vectors, play an important role.