High-dimensional linear regression has been intensively studied in the community of statistics in the last two decades. For convenience of theoretical analyses, classical methods usually assume independent observations and subGaussian-tailed errors. However, neither of them hold in many real high-dimensional time-series data. Recently [Sun, Zhou, Fan, 2019, J. Amer. Stat. Assoc., in press] proposed Adaptive Huber Regression (AHR) to address the issue of heavy-tailed errors. They discover that the robustification parameter of the Huber loss should adapt to the sample size, the dimensionality and (1+\delta) -moments of the heavy-tailed errors. We progress in a vertical direction and justify AHR on dependent observations. Specially, we consider an important dependence structure---Markov dependence. Our results show that the Markov dependence impacts on the adaption of the robustification parameter and the estimation of regression coefficients in the way that the sample size should be discounted by a factor depending on the spectral gap of the underlying Markov chain.
The main purpose of this workshop was to assemble international leaders from statistics and machine learning to identify important research problems, to cross-fertilize between the disciplines, and to ultimately start coordinated research efforts toward better solutions. The workshop focused on discussing modern methods for analysis complex high dimensional data with applications to econometrics, finance, biomedicine, genomics etc.
Abstract Markowitz (1952, 1959) laid down the ground-breaking work on mean-variance analysis without gross exposure constraints. Under this framework, the theoretical optimal allocation vector can be different from the estimated one due to intrinsic difficulty of estimating a large covariance matrix and return vector. This can result in adverse performance in portfolio selected based on empirical data due to noise accumulation on estimation errors (Jagannathan and Ma, 2003; Fan, Fan and Lv, 2008). We address this problem by introducing the gross-exposure constrained mean-variance portfolio selection. We show that with gross-exposure constraint the theoretical optimal portfolios have similar performance as empirically selected ones based on estimated covariance matrices and there is no noise accumulation effect from estimation of covariance matrices. This gives theoretical justification to the empirical results
Statistics Theory and Methodsmathscidoc:1912.43433
Princeton University, USA and Hong Kong University of Science and Technology, HKSAR Extent and terms of a thesis are specified in directions for its elaboration that are opened to the public on the web sites of the faculty