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Under standard assumptions including stationary and serially uncorrelatedGaussian gravitational wave stochastic background signal and noisedistributions, as well as homogenous detector sensitivities, the standardcross-correlation detection statistic is known to be optimal in the sense ofminimizing the probability of a false dismissal at a fixed value of theprobability of a false alarm. The focus of this paper is to analyze thecomparative efficiency of this statistic, versus a simple alternative statisticobtained by cross-correlating the \textit{squared} measurements, in situationsthat deviate from such standard assumptions. We find that differences indetector sensitivities have a large impact on the comparative efficiency of thecross-correlation detection statistic, which is dominated by the alternativestatistic when these differences reach one order of magnitude. This effectholds even when both the signal and noise distributions are Gaussian. While thepresence of non-Gaussian signals has no material impact for reasonableparameter values, the relative inefficiency of the cross-correlation statisticis less prominent for fat-tailed noise distributions but it is magnified incase noise distributions have skewness parameters of opposite signs. Ourresults suggest that introducing an alternative detection statistic can lead tonoticeable sensitivity gains when noise distributions are possibly non-Gaussianand/or when detector sensitivities exhibit substantial differences, a situationthat is expected to hold in joint detections from Advanced LIGO and AdvancedVirgo, in particular in the early phases of development of the detectors, or injoint detections from Advanced LIGO and Einstein Telescope.

On the (In)Efficiency of the Cross-Correlation Statistic for  Gravitational Wave Stochastic Background Signals with Non-Gaussian Noise and  Heterogeneous Detector Sensitivities