## Some Bayes Risk Consistent Non-parametric Methods for Classification |

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25 Range Median 500 Samples 9907 Mixture Ro absolutely continuous algorithm An(x assigned to tt Bayes risk consistent Bn(x Chapter CLASSIFICATION METHOD Classification Rule Based correct classification denote Duda and Hart empiric distribution function empiric function empiric rules error rate estimated non-error rates estimated probability extrema locations extrema of H extremum finite number Glick 24 Hart 15 Hence histogram method Histogram Rule HR Hn(r Hn(x Hn(Yj hQ(n identically distributed integer Lemma Let kn locations of H mean non-error rates Mean S.d. Mean non-decreasing NON-PARAMETRIC METHODS nonparametric density estimation number of extrema number of observations Optimal Classification optimal rule order statistics probability of correct Proof random variable Rate of 500 relative extrema relative supremum Ryzin 56 S.d. Mean S.d. sup Xo sup|H supremum or infimum test sample Theorem IV.4.1 thesis training sample univariate University of Wisconsin-Madison x-j is fixed