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Standard spatial time-frequency distribution (STFD) estimators, derived based on the Gaussian noise assumption, are known to have poor performance in the case of impulsive noise. Recently, different STFD estimators have been proposed, which, based on simulations, are claimed to be robust. In this paper, we provide an influence function robustness analysis of STFD estimators. We derive the influence functions for the asymptotic and for the finite-sample case and study robustness of the standard, as well as for some recently proposed robust STFD estimators. The empirical influence function gives practitioners a simple way to pre-select STFD estimators for their scenario. Our analysis confirms that, unlike for the standard estimator, the proposed robust estimators yield a bounded influence function and are robust over a broad class of distributions. Future research on STFD estimation will allow for the design of robust and efficient estimators based on the influence function.