## 1 Introduction  
The ocean has become an underdeveloped high-tech resource, and it's development has always been the strategic goal of all countries [1, 2]. The autonomous underwater vehicle (AUV) plays an indispensable role in surveying, underwater reconnaissance, and underwater monitoring in the ocean. Positioning and navigation methods with higher accuracy are necessities for the AUV, as are key technologies to ensure a safe return and safe underwater work [3, 4]. For one thing, there are all kinds of noise interferences in the ocean, such as salt cliffs, ocean currents, and so on, so it is hard to make an accurate underwater noise model [5]. Moreover, plenty of navigation methods based on electromagnetic propagation cannot be employed because of this signal rapidly decays underwater [6]. Additionally, the positioning error accumulation of the micro-electro-mechanical System (MEMS) [7, 8, 9, 10] is very serious with time. Those above shortcomings will lead to bigger measurement errors, which have negative effects on AUV positioning [11]. Hence, it is important that estimation algorithms work as they are intended for AUVs [12].  
The extended Kalman filter (EKF) is employed in the nonlinear system by means of the linearization approach. However, the truncated error is introduced in the nonlinear model when using the Taylor series expansion [13, 14, 15, 16, 17, 18, 19]. When the nonlinearity of system is extremely strong, the filter performs poorly because of the truncated error. Hence, the problem of the truncated error is solved by using the unscented transformation in the unscented Kalman filter (UKF) [20, 21, 22]. The UKF is more suitable than EKF for MEMS grade inertial navigation systems (INS) [23, 24, 25]. Additionally, it is difficult for INS to describe precise noise information due to the fact that the performance of the sensors varies in the underwater environment [26, 27]. The expectation-maximization (EM) method has the advantage of simplicity, and each iteration of the EM method increases the similarity until a suboptimal solution is achieved [28; 29; 30]. The EM method is different from other traditional optimization methods, and it is not necessary to calculate a partial derivative. Moreover, it is not essential to be concerned about setting the iteration number [31; 32; 33; 34; 35; 36]. Therefore, the EM method is an ideal adaptive method to estimate precise noise information.  
A novel statistical regression adaptive Kalman filtering (SRAKF) is proposed by making use of the EM method and UKF so that the state vector of the AUV is estimated more accurately by UKF, and the EM method can help to calculate the suboptimal solution of the unknown parameters in the state-space model. The proposed SRAKF in this paper has better robustness and higher estimation accuracy, especially in the unknown and complex underwater environment.  
The novelties are shown as follows:  
1. The average absolute error is used as the criterion discard or replacement, so the data is smoothed by the combination of mean filtering and linear smoothing;
2. The positioning accuracy is improved by SRAKF that can calculate the accurate solutions of the predicted error covariance matrix and measurement noise matrix;
3. The unscented transformation is used to calculate the nonlinear integral operation. This method is conductive to increasing the state estimation accuracy by suppressing the truncation error caused by numerical integration.  
The structure of the paper is shown below. The underwater positioning error model is described and presents a method to reject the outlier data in Section 2. In Section 3, the new proposed algorithm is described in detail, and the convergence property of the SRAKF is proved. Section 4 uses real experiments to show a dramatic advance in the SRAKF's performance over UKF and CKF. The main conclusions are summarized in the last section.