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Donoho and Kipnis (2022) showed that the the higher criticism (HC) teststatistic has a non-Gaussian phase transition but remarked that it is probablynot optimal, in the detection of sparse differences between two large frequencytables when the counts are low. The setting can be considered to beheterogeneous, with cells containing larger total counts more able to detectsmaller differences. We provide a general study here of sparse detectionarising from such heterogeneous settings, and showed that optimality of the HCtest statistic requires thresholding, for example in the case of frequencytable comparison, to restrict to p-values of cells with total counts exceedinga threshold. The use of thresholding also leads to optimality of the HC teststatistic when it is applied on the sparse Poisson means model of Arias-Castroand Wang (2015). The phase transitions we consider here are non-Gaussian, andinvolve an interplay between the rate functions of the response and sample sizedistributions. We also showed, both theoretically and in a numerical study,that applying thresholding to the Bonferroni test statistic results in bettersparse mixture detection in heterogeneous settings.

Thresholding the higher criticism test statistics for optimality in a heterogeneous setting