Test for high-dimensional outliers with principal component analysis

We herein consider a test of outlier detection for high-dimensional, low-sample-size (HDLSS) data. Although outlier detection is a fundamental problem, it has not been extensively studied in the HDLSS setting. We derive asymptotic properties of the first principal component scores with outliers. We consider high-dimensional outlier detection by applying the asymptotic properties to the Grubbs test, a well-known method for testing outliers. Our results indicate that the test statistic provides preferable performance for both the size and power. Using this test procedure, we propose an algorithm to identify multiple outliers. We present an investigation of the theoretical properties of a sure independent screening and it can achieve complete identification of the outliers with high accuracy. Finally, we investigate the performance for both numerical studies and real data analyses as compared to available outlier detection methods in HDLSS settings. The proposed method exhibits superiority in terms of not only correctly detecting outliers, but also identifying a number of false identifications.