  Under standard assumptions including stationary and serially uncorrelated
Gaussian gravitational wave stochastic background signal and noise
distributions, as well as homogenous detector sensitivities, the standard
cross-correlation detection statistic is known to be optimal in the sense of
minimizing the probability of a false dismissal at a fixed value of the
probability of a false alarm. The focus of this paper is to analyze the
comparative efficiency of this statistic, versus a simple alternative statistic
obtained by cross-correlating the \textit{squared} measurements, in situations
that deviate from such standard assumptions. We find that differences in
detector sensitivities have a large impact on the comparative efficiency of the
cross-correlation detection statistic, which is dominated by the alternative
statistic when these differences reach one order of magnitude. This effect
holds even when both the signal and noise distributions are Gaussian. While the
presence of non-Gaussian signals has no material impact for reasonable
parameter values, the relative inefficiency of the cross-correlation statistic
is less prominent for fat-tailed noise distributions but it is magnified in
case noise distributions have skewness parameters of opposite signs. Our
results suggest that introducing an alternative detection statistic can lead to
noticeable sensitivity gains when noise distributions are possibly non-Gaussian
and/or when detector sensitivities exhibit substantial differences, a situation
that is expected to hold in joint detections from Advanced LIGO and Advanced
Virgo, in particular in the early phases of development of the detectors, or in
joint detections from Advanced LIGO and Einstein Telescope.
