A Higher Order Analysis of Sparse Normal Mixtures

The Donoho and Jin (2004) higher criticism statistic (HC) is an increasingly popular tool in sparse mixture detection, feature selection, and classification problems. We consider the extensively studied sparse normal mixture detection problem, which has been used as a testing ground for other procedures (such as the Jager and Wellner (2007) phi-divergence statistics, of which HC is a particular case). We reveal an alternative interpretation for HC, which perhaps explains some of its strengths and weaknesses in sparse mixture detection. Several implications of this interpretation are established in the sparse normal mixture detection problem. In motivating their statistic HC, Donoho and Jin (2004) state that `it is not clear that [the generalised likelihood ratio test (GLRT)] can be relied on to detect subtle departures from the null hypothesis'. We will demonstrate that the GLRT can be relied on to detect the subtle departures in two sparse normal mixture problems. We show that there are sequences of local alternatives where the GLRT has limiting power one, but the Jager and Wellner (2007) procedures (including HC) have no limiting power