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In this letter, a selective kernel principal component analysis (KPCA) algorithm based on high-order statistics is proposed for anomaly detection in hyperspectral imagery. First, KPCA is performed on the original hyperspectral data to fully mine the high-order correlation between spectral bands. Then, the average local singularity (LS) is defined based on the high-order statistics in the local sliding window, which is used as a measure for selecting the most informative nonlinear component for anomaly detection. By the selective KPCA, information on anomalous targets is extracted to maximum extent, and background clutters are well suppressed in the selected component. Finally, the selected component with maximum average LS is used as input for anomaly detectors. Numerical experiments are conducted on real hyperspectral images collected by the airborne visible/infrared imaging spectrometer. The results strongly prove the effectiveness of the proposed algorithm.