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The optimal performance of the Kalman filters is highly dependent on the measurement and process noise characteristics, making the whole system unable to achieve the desired estimation in the presence of non-Gaussian mean noise distribution and high initial uncertainties. Recently, the H-infinity filter, as a robust algorithm, has been broadly used, as it is not being dependent on the pre-knowledge of the noise nature; however, making a balance between high robustness and estimation accuracy is a challenging issue. Hence, to overcome this problem, a new adaptive H-infinity extended Kalman filter (AHEKF) was designed in this paper, which benefits from both high robustness and precision. The suggested algorithm contains two adaptive sections to achieve high accuracy as well as controlling the effects of time-varying noise characteristics, high initial uncertainties, and abnormal data that can degrade the accuracy of state estimation in an integrated navigation system. The presented algorithm was used to integrate data from two independent sensors data. The simulation results for an inertial navigation system (INS)/global positioning system (GPS) sensor fusion are presented and compared with the standard H-infinity filter, extended Kalman filter (EKF), and unscented Kalman filter (UKF) to show the effectiveness of the proposed algorithm. Evaluations demonstrate that the AHEKF achieves over 50% higher accuracy and robustness, and over 2.5 times faster convergence of estimation errors than the standard H-infinity filter.