Title: Robust mean and eigenvalues regularized covariance matrix estimation
Authors: Clifford Lam - London School of Economics and Political Science (United Kingdom)
Wenyu Cheng - London School of Economics and Political Science (United Kingdom) [presenting]
Abstract: Covariance matrix is a common tool for summarising linear relationships between variables. The topic is particularly of interest in the high-dimensional setting, where the classical sample covariance estimator is no longer optimal. Non-parametric eigenvalue shrinkage covariance estimator (NERCOME) offers one solution by shrinking extreme eigenvalues non-linearly. It makes no structural assumptions on the covariance matrix and guarantees a positive definite outcome. However, its good performance relies on the availability of twelve moments, which is hard to verify and often unsatisfied in the real world. The unpredictable presence of few outliers from highly asymmetric or fat-tailed distributions could significantly jeopardise its performance. We draw on recent developments from the robust statistics literature. Specifically, we focus on generalising Catoni loss function to alleviate the impacts of extreme observations. The improved influence function, now requiring the existence of just over one moment, produces narrower bounds on estimated means. Incorporating these findings, the robust NERCOME behaves consistently across different distributional settings, while maintaining overall estimation efficiency and other desirable properties as in NERCOME. We challenge the robust NERCOME with highly skewed and leptokurtic scenarios through extensive simulation studies. Applications in financial data are provided in the end.