An inexpensive Microelectromechanical systems (MEMS) gyroscope smartphone sensor in a smartphone is considered for angular velocity estimation. The performance of the MEMS gyroscope is substandard due to random errors associated with it. Kalman and adaptive Kalman filters are applied to improve the performance of the MEMS gyroscope estimation. In this paper, a Sage-Husa adaptive Kalman filter (SHAKF) is extented to residual-based estimation of the measurement noise covariance matrix to ensure its positive definiteness. It is complemented by an innovation-based estimation of the process noise covariance matrix to provide numerical stability of the resulting SHAKF. In addition, a Mahalanobis distance-based judging index is proposed within the framework of the SHAKF to reduce the impact of the outliers on the recursive estimation of the angular velocity estimates. Estimation results obtained from measured smartphone data and simulations reveal that the proposed weighted robust Sage-Husa adaptive Kalman filter (WRSHAKF) clearly outperforms the SHAKF in terms of the achievable Allan and standard deviations with low computational complexity.