In this paper, we consider the problem of determining the presence of a given signal in a high-dimensional observation with unknown covariance matrix by using an adaptive matched filter. Traditionally such filters are formed from the sample covariance matrix of some given training data, but, as is well-known, the performance of such filters is poor when the number of training data $n$ is not much larger than the data dimension $p$. We thus seek a covariance estimator to replace sample covariance. To account for the fact that $n$ and $p$ may be of comparable size, we adopt the "large-dimensional asymptotic model" in which $n$ and $p$ go to infinity in a fixed ratio. Under this assumption, we identify a covariance estimator that is asymptotically detection-theoretic optimal within a general shrinkage class inspired by C. Stein, and we give consistent estimates for conditional false-alarm and detection rate of the corresponding adaptive matched filter.
翻译:在本文中,我们考虑了使用适应性匹配过滤器确定某一信号在高维观测中存在并具有未知共差矩阵的问题。传统上,这种过滤器是由某些特定培训数据的样本共差矩阵组成的,但众所周知,当培训数据数量不比数据维度大得多时,这种过滤器的性能就差了。因此,我们寻求一个共差估计器来取代样本共差。考虑到美元和美元可能具有类似规模的事实,我们采用了“大维无损模型”,其中,美元和美元将固定比例地用于无限化。在此假设下,我们确定一个在C. Stein所启发的一般缩水层中,不同时进行检测和测得最佳的共差估计器,我们对于相应的适应匹配过滤器的有条件假武器率和检测率给出一致的估计。