A class of robust Cubature Kalman filter(CKF) algorithm with statistical regression is proposed to solve the problem that the conventional CKF declines in accuracy and further diverges when the noise is not Gaussian noise or its prior statistic is unknown. Two kinds of robust CKFs with linear approximation regression or not are deduced and filtering steps are designed. The directly nonlinear regression overcomes the shortcoming of CKF with linear approximation of measurement align. Simulation example with a model of mixed Gaussian noise analyzes and contrasts the performances of filter with the three kinds of Cubature Kalman Filter. The results show that the two robust Cubature Kalman filters outbalance the conventional CKF in the accuracy and consistency of filtering, and the robust CKF without linear approximation owns stronger robustness and better performance compared with the other robust CKF.