## 1 Introduction  
Traditional adaptive filtering builds cost functions based on the minimum mean square error (MMSE) criterion, which is desirable under the assumption that the system noise follows a Gaussian distribution [1; 2]. However, when the system noise contains impulsive components, the performance of the MMSE criterion is severely degraded [3]. To solve the problem and reduce the impact of impulsive noise, robust adaptive filtering has been studied, adopting criteria beyond MMSE to construct robust cost functions, such as M-estimation statistics [4; 5], least mean p-power [6; 7], etc. In recent years, inspired by information learning theory [8], the maximum correntropy criterion (MCC) and its extensions have been studied extensively and considered to be effective when dealing with non-Gaussian system noise [9; 10; 11; 12; 13]. Correntropy in the MCC is a measure of the distance between two random variables based on a Gaussian kernel function. The distance can be also called correntropy induced metric (CIM) in the input space and CIM behaves like different norms from L2-norm to L1-norm depending on the distance between the two inputs [14; 15]. With the above property, correntropy can stay insensitive to data outliers when set with a small kernel bandwidth,which diminishes the influence of impulsive noise in adaptive filtering and ensures robustness. One classic extension of the MCC is the generalized maximum correntropy criterion (GMCC) based on a generalized Gaussian kernel, which adds a shape parameter to the Gaussian kernel and improves the adaptability faced with different kinds of non-Gaussian system noises [11; 12; 13]. The MCC and GMCC have been extensively studied and applied to many applications in system parameter identification and state estimation, such as channel estimation [16; 17; 18], blind multiuser detection [19], SINS/GPS-integrated system estimation [20], spacecraft relative state estimation [21], acoustic echo cancellation [22], etc. The above criteria are based on a symmetric Gaussian kernel and are desirable when dealing with symmetric distributed system noises. However, an asymmetric signal or noise exists in many areas of data analysis and signal processing, such as in insurance analysis [23], finance analysis [24; 25], image processing [26], etc. Under an asymmetric noise environment, the estimation error follows a skewed distribution, thus the MCC and GMCC based on a symmetric Gaussian kernel are no longer suitable. In [27], a very enlightening idea was proposed where a maximum asymmetric correntropy criterion (MACC) was obtained by replacing the Gaussian kernel in the MCC with an asymmetric Gaussian kernel. An asymmetric Gaussian kernel can process positive and negative estimation errors with different kernel widths, respectively, so that it can better adapt to asymmetric noise. Similar to the expansion from the MCC to the GMCC [11], this paper adds a shape parameter to MACC leading to a generalized maximum asymmetric correntropy criterion (GMACC), which is based on a generalized asymmetric Gaussian (GAG) kernel [28; 29; 30; 31] and is more flexible than MACC. Ref. [28] used finite asymmetric generalized Gaussian mixture model learning to improve the effect of infrared object detection. Ref. [29] presented an image segmentation method based on a parametric and unsupervised histogram, which was assumed to be a mixture of asymmetric generalized Gaussian distributions. In [31], an asymmetric generalized Gaussian density was used to deal with the asymmetric shape property of certain texture classes. Ref. [30] expanded the concept of a generalized Gaussian distribution to include asymmetry so as to model distributions of different tail lengths as well as different degrees of asymmetry. The above research works on the GAG kernel mostly used the kernel function to build data models, while this paper is inspired by the GMACC to construct the cost function of robust adaptive filtering, so as to improve the research framework as an extension of MACC-based adaptive filtering.