Consider the problem of simultaneous estimation of location and variance matrix under Huber's contaminated Gaussian model. First, we study minimum $f$-divergence estimation at the population level, corresponding to a generative adversarial method with a nonparametric discriminator and establish conditions on $f$-divergences which lead to robust estimation, similarly to robustness of minimum distance estimation. More importantly, we develop tractable adversarial algorithms with simple spline discriminators, which can be implemented via nested optimization such that the discriminator parameters can be fully updated by maximizing a concave objective function given the current generator. The proposed methods are shown to achieve minimax optimal rates or near-optimal rates depending on the $f$-divergence and the penalty used. We present simulation studies to demonstrate advantages of the proposed methods over classic robust estimators, pairwise methods, and a generative adversarial method with neural network discriminators.
翻译:考虑在Huber被污染的高斯模式下同时估计位置和差异矩阵的问题。 首先,我们研究人口层面的最低美元差异值估算,对应非参数歧视者的基因对抗性对抗方法,并针对导致可靠估算的美元差异值设定条件,类似于最低距离估算的稳健性。 更重要的是,我们开发了使用简单样条区别器的可移动对抗算法,可通过嵌套优化实施,这样,根据当前生成器,通过最大限度地实现一个凝固目标功能,可充分更新歧视参数。 显示拟议方法可达到最低最佳比率或近于最佳比率,取决于所使用的美元差异值和罚款。 我们提出模拟研究,以展示拟议方法优于传统的稳健测算器的优势,配对式方法,以及与神经网络歧视器的基因化对抗方法。