For vehicle state estimation, conventional Kalman filters work well under Gaussian assumptions. Still, they are likely to degrade dramatically in the practical non-Gaussian situation (especially the noise is heavy-tailed), showing poor accuracy and robustness. This article presents an estimation technique based on the Maximum Correntropy Criterion (MCC) combined with an adaptive extended Kalman filter (AEKF), and an extended Kalman filter (EKF) based on the MCC has also been studied. A lateral-longitudinal coupled vehicle model is developed, while an observer containing the state vectors such as yaw rate, sideslip angle, vehicle velocity and tire cornering stiffness is designed using easily available in-vehicle sensors and low-cost GPS. After analyzing the algorithmic complexity, the proposed algorithm is validated by Sine Steering Input and Double Lane Change driving scenarios. Finally, it is found that MCC combined with AEKF/EKF (MCAEKF/MCEKF) has stronger robustness and better estimation accuracy than AEKF/EKF in dealing with non-Gaussian noise for vehicle state estimation.