ARE for Testing; Convergence Rate of Kernel Density Estimation

The t-test is as defined in the previous lecture. It has slope 1/σ. The Mann-Whitney test rejects if 1 nm ∑ i,j I(Xi ≤ Yj) is large. Note. The Mann-Whitney statistic has a relationship with the area under the ROC curve (AUC) for classification algorithms with a tunable parameter. The ROC plot has one axis for proportion of false positives and one axis for proportion of true positives; as we move to the right, the classifier becomes more sensitive to true positives, but misclassifies more negative points as positive. Nearly-flat ROC curves are bad and strongly humped ROC curves are good (see Figure 1). The empirical ROC curve shows the classifier’s performance on the training data and has the form of discrete stair-steps. If we consider the positive examples to be one population Y1, . . . , Ym and the negative examples to be another population X1, . . . , Xn, and consider