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Likelihood-based test statistics for the task of detecting a radioactive source in background using a gamma-ray imaging system often have intractable distributions. This complicates the tasks of predicting detection performance and setting thresholds that ensure desired false-alarm rates. Asymptotic distributions of test statistics can aid in predicting performance and in setting detection thresholds. However, in applications with complex sensors, like gamma-ray imaging, often only approximate statistical models for the measurements are available. Standard asymptotic approximations can yield inaccurate performance predictions when based on misspecified models. This paper considers asymptotic properties of detection tests based on maximum likelihood (ML) estimates under model mismatch, i.e., when the statistical model used for detection differs from the true distribution. We provide general expressions for the asymptotic distribution of likelihood-based test statistics when the number of measurements is Poisson, and expressions specific to gamma-ray source detection that one can evaluate using a modest amount of data from a real system or Monte Carlo simulation. Considering a simulated Compton imaging system, we show that the proposed expressions yield more accurate detection performance predictions than previous expressions that ignore model mismatch. These expressions require less data and computation than conventional empirical methods.