Multiple testing of local maxima for detection of peaks on the (celestial) sphere

Abstract

We present a topological multiple testing scheme for detecting peaks on the sphere under isotropic Gaussian noise, where tests are performed at local maxima of the observed field filtered by the spherical needlet transform. Our setting is different from the standard Euclidean large domain asymptotic framework, yet highly relevant to realistic experimental circumstances for some important areas of application in astronomy, namely point-source detection in cosmic Microwave Background radiation (CMB) data. Motivated by this application, we shall focus on cases where a single realization of a smooth isotropic Gaussian random field on the sphere is observed, and a number of well-localized signals are superimposed on such background field. The proposed algorithms, combined with the Benjamini–Hochberg procedure for thresholding p-values, provide asymptotic control of the False Discovery Rate (FDR) and power consistency as the signal strength and the frequency of the needlet transform get large.