广义鲁棒收缩估计及其在STAP检测问题中的应用
Generalized robust shrinkage estimator and its application to STAP detection problem
摘要：
最近，在协方差阵估计的背景下，为了改进和正则化泰勒估计量[1]（也称为固定点估计量（FPE）[2]）的性能，在[3]中引入了“收缩”固定点估计量。首先，本文推广了文[3，4]的结果，给出了“收缩”不动点算法的一般解。其次，通过对广义鲁棒收缩估计的分析，证明了当收缩参数$β$（损失因子）趋于0时，广义鲁棒收缩估计收敛于唯一解。这个解正好是FPE，其逆的迹等于问题的维数。这个一般的结果允许一个给另一种解释的FPE和更一般地说，在极大似然方法的协方差阵估计时，添加约束。然后，一些模拟说明我们的理论结果以及如何选择一个最佳的收缩因子。最后，这项工作被应用到一个空时自适应处理（STAP）检测问题的实际STAP数据。
英文摘要：
Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tyler's estimator [1] also called the Fixed-Point Estimator (FPE) [2], a "shrinkage" fixed-point estimator has been introduced in [3]. First, this work extends the results of [3,4] by giving the general solution of the "shrinkage" fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter $β$ (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data.
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