## 1 Introduction  
In statistical signal processing applications, such as radar and communications, the sample covariance matrix plays an essential role [1]. It is usually estimated from \(N\) sample data vectors \([\mathbf{x}_{0}\ \mathbf{x}_{1}\ \ldots,\mathbf{x}_{N-1}]\), where \(\mathbf{x}_{n}\) is assumed to be \(L\times 1\) identical and independently distributed (IID). The maximum likelihood covariance matrix estimate is [2]  
\[\hat{\mathbf{C}}=\frac{1}{N}\sum_{n=0}^{N-1}\mathbf{x}_{n}\mathbf{x}_{n}^{H}, \tag{1}\]  
where \(H\) denotes a Hermitian. A good covariance matrix estimate usually requires the number of samples \(N\) to be sufficiently large. For instance, in space-time adaptive processing (STAP), it requires \(N\geq 2L\) to have a good clutter covariance matrix estimate of size \(L\times L\)[3]. In practice, however, this is not valid due to the nonstationary environment. For example, the data for a STAP system are often nonstationary due to the heterogeneous clutter [1]. The number of data that are sufficiently IID (homogeneous) can be relatively small \(N\leq L\)[3]. Techniques such as thresholding and banding are common ways to achieve better covariance matrix estimation. The thresholding method sets small elements of the sample covariance matrix to zero to obtain better estimators [4, 5]. Another approach is to band or taper the sample covariance matrix [6, 7]. [PERSON] et al. [8] proposed a Cholesky-based covariance regularization method to ensure positive definiteness. In practice, the inverse covariance matrix may be of primary interest. When the data are multivariate Gaussian, the inverse of the covariance matrix can be used to infer the conditional dependence structure of random variables [9]. Several researchers have investigated different methods for banding the inverse of the covariance matrix. [PERSON] et al. proposed estimating the covariance matrix by banding the Cholesky-factor matrix and applying kernel smoothing estimation [10]. [PERSON] demonstrated that within the bandable class of covariance matrices, the estimator \(\hat{\mathbf{C}}^{-1}\) obtained by banding the Cholesky-factor matrix of the covariance matrix's inverse is consistent [4; 6]. [PERSON] et al. explored adaptive banding covariance estimation for high-dimensional multivariate longitudinal data [11]. However, not much work is available to provide a criterion for deciding if a covariance matrix is bandable. Such a criterion would be useful for deciding if the banding technique is a suitable strategy in covariance matrix estimation tasks. Moreover, other covariance estimation methods, such as modeling the covariance matrix as a time-varying autoregressive moving average (ARMA) model [12], also require one to test if the model has a good fit, which is similar to but distinct from model order estimation techniques such as the minimum description length (MDL), AIC, BIC, and Bayesian exponential embedded family [13]. Some recent tests for bandedness can be found in [14], where a method for estimating matrix bandwidth is presented. [PERSON] et al. developed several tests for sparse high-dimensional covariance matrices [15]. In [9], [PERSON] et al. proposed test statistics for detecting band size and applied them to cancer data analysis. In contrast to these works, we pose the problem as a classical parameter hypothesis testing problem, which allows us to employ well-established detection theorems and algorithms in statistical signal processing. In many practical fields, such as radar and communications, the data and parameters are often complex-valued [16]; therefore, we consider the bandedness testing problem for a complex-valued covariance matrix herein. Some general topics related to complex-valued signal processing can be found in [17; 18]. In [19], [PERSON] and [PERSON] derived the complex parameter Rao test, which allows one to develop a Rao test for complex parameters in a complex-valued domain directly. Based on [19], [PERSON] et al.