High-dimensional covariance matrix estimation with applications in finance and genomic studies

Résumé

We consider the problem of estimating a high-dimensional (HD) covariance matrix when the sample size is smaller, or not much larger, than the dimensionality of the data, which could potentially be very large. We develop a regularized sample covariance matrix (RSCM) estimator that is based on consistent estimators of the unknown optimal (oracle) shrinkage parameters that yield the minimum mean squared error (MMSE) between the RSCM and the true covariance matrix when the data is sampled from an unspecified elliptically symmetric distribution. The proposed covariance estimator is then used in portfolio optimization problems in finance and microarray data analysis (MDA). In portfolio optimization problem we use our estimator for optimally allocating the total wealth to a large number of assets, where optimality means that the risk (i.e., variance of portfolio returns) is minimized. Microarray technology is a powerful approach for genomics research that allows monitoring the expression levels of tens of thousands of genes simultaneously. We develop a compressive regularized discriminant analysis (CRDA) method based on our covariance estimator and illustrate its effectiveness in MDA. Our analysis results on real stock market data and microarray data illustrate that the proposed approach is able to outperform the current benchmark methods. BIO: Esa Ollila received the M.Sc. degree in mathematics from the University of Oulu, in 1998, Ph.D. degree in statistics with honors from the University of Jyvaskyla, in 2002, and the D.Sc. (Tech) degree with honors in signal processing from Aalto University, in 2010. From 2004 to 2007 he was a post-doctoral fellow and from August 2010 to May 2015 an Academy Research Fellow of the Academy of Finland. The academic year 2010-2011 he spent as a Visiting Post-doctoral Research Associate with the Department of Electrical Engineering, Princeton University, Princeton, NJ. Currently, since June 2015, he is an Associate Professor of Signal Processing at Aalto University. He is also an adjunct Professor (statistics) of Oulu University. His research interests focus on theory and methods of statistical signal processing, multivariate statistics and data science.