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Nowadays, automatic modulation classification (AMC) plays an important role in both cooperative and non-cooperative communication applications. Very often, multipath fading channels result in the severe AMC performance degradation or induce large classification errors. The negative impacts of multipath fading channels on AMC have been discussed in the existing literature but no solution has ever been proposed so far to the best of our knowledge. In this paper, we propose a new robust AMC algorithm, which applies higher-order statistics (HOS) in a generic framework for blind channel estimation and pattern recognition. We also derive the Cramer-Rao lower bound for the fourth-order cumulant estimator when the AMC candidates are BPSK and QPSK over the additive white Gaussian noise channel, and it is a nearly minimum-variance estimator leading to robust AMC features in a wide variety of signal-to-noise ratios. The advantage of our new algorithm is that, by carefully designing the essential features needed for AMC, we do not really have to acquire the complete channel information and therefore it can be feasible without any a priori information in practice. The Monte Carlo simulation results show that our new AMC algorithm can achieve the much better classification accuracy than the existing AMC techniques.