论文标题
对数似然比的渐近正态性和弱检测的基本限制
Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices
论文作者
论文摘要
我们考虑在排名一的尖峰模型中检测信号的存在的问题。对于一般的非高斯噪声，假设信号是从rademacher先验中得出的，我们证明，当信号噪声比率低于一定阈值时，尖峰模型的对数可能性比（LR）会收敛到高斯。阈值是最佳的，因为在其上方，可以通过转换的主组件分析（PCA）进行可靠的检测。从log-lr的限制高斯的平均值和方差，我们计算了I型误差和类型II误差的限制。对于噪声不对称，但信号是对称的，我们还证明了一个排名一的尖刺的IID模型的结果相似。
We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. For general non-Gaussian noise, assuming that the signal is drawn from the Rademacher prior, we prove that the log likelihood ratio (LR) of the spiked model against the null model converges to a Gaussian when the signal-to-noise ratio is below a certain threshold. The threshold is optimal in the sense that the reliable detection is possible by a transformed principal component analysis (PCA) above it. From the mean and the variance of the limiting Gaussian for the log-LR, we compute the limit of the sum of the Type-I error and the Type-II error of the likelihood ratio test. We also prove similar results for a rank-one spiked IID model where the noise is asymmetric but the signal is symmetric.
