Matched, mismatched, and robust scatter matrix estimation and hypothesis testing in complex t-distributed data

Scatter matrix estimation and hypothesis testing are fundamental inference problems in a wide variety of signal processing applications. In this paper, we investigate and compare the matched, mismatched, and robust approaches to solve these problems in the context of the complex elliptically symmetric (CES) distributions. The matched approach is when the estimation and detection algorithms are tailored on the correct data distribution, whereas the mismatched approach refers to the case when the scatter matrix estimator and the decision rule are derived under a model assumption that is not correct. The robust approach aims at providing good estimation and detection performance, even if suboptimal, over a large set of possible data models, irrespective of the actual data distribution. Specifically, due to its central importance in both the statistical and engineering applications, we assume for the input data a complex t-distribution. We analyze scatter matrix estimators derived under the three different approaches and compare their mean square error (MSE) with the constrained Cramér-Rao bound (CCRB) and the constrained misspecified Cramér-Rao bound (CMCRB). In addition, the detection performance and false alarm rate (FAR) of the various detection algorithms are compared with that of the clairvoyant optimum detector.