Non-Gaussian noises always exist in the nonlinear system, which usually lead to inconsistency and divergence of the regression and identification applications. The conventional evolving fuzzy systems (EFSs) in common sense have succeeded to conquer the uncertainties and external disturbance employing the specific variable structure characteristic. However, non-Gaussian noises would trigger the frequent changes of structure under the transient criteria, which severely degrades performance. Statistical criterion provides an informed choice of the strategies of the structure evolution, utilizing the approximation uncertainty as the observation of model sufficiency. The approximation uncertainty can be always decomposed into model uncertainty term and noise term, and is suitable for the non-Gaussian noise condition, especially relaxing the traditional Gaussian assumption. In this paper, a novel incremental statistical evolving fuzzy inference system (SEFIS) is proposed, which has the capacity of updating the system parameters, and evolving the structure components to integrate new knowledge in the new process characteristic, system behavior, and operating conditions with non-Gaussian noises. The system generates a new rule based on the statistical model sufficiency which gives so insight into whether models are reliable and their approximations can be trusted. The nearest rule presents the inactive rule under the current data stream and further would be deleted without losing any information and accuracy of the subsequent trained models when the model sufficiency is satisfied. In our work, an adaptive maximum correntropy extend Kalman filter (AMCEKF) is derived to update the parameters of the evolving rules to cope with the non-Gaussian noises problems to further improve the robustness of parameter updating process. The parameter updating process shares an estimate of the uncertainty with the criteria of the structure evolving process to make the computation less of a burden dramatically. The simulation studies show that the proposed SEFIS has faster learning speed and is more accurate than the existing evolving fuzzy systems (EFSs) in the case of noise-free and noisy conditions.