We propose a robust portfolio optimization approach based on quantile statistics. The proposed method is robust to extreme events in asset returns, and accommodates large portfolios under limited historical data. Specifically, we show that the risk of the estimated portfolio converges to the oracle optimal risk with parametric rate under weakly dependent asset returns. The theory does not rely on higher order moment assumptions, thus allowing for heavy-tailed asset returns. Moreover, the rate of convergence quantifies that the size of the portfolio under management is allowed to scale exponentially with the sample size of the historical data. The empirical effectiveness of the proposed method is demonstrated under both synthetic and real stock data. Our work extends existing ones by achieving robustness in high dimensions, and by allowing serial dependence.

In this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions. We assume that the data are collected from n subjects, each of which consists of T possibly dependent observations. The graphical models of subjects vary, but are assumed to change smoothly corresponding to a measure of closeness between subjects. We propose a kernel based method for jointly estimating all graphical models. Theoretically, under a double asymptotic framework, where both (T,n) and the dimension d can increase, we provide the explicit rate of convergence in parameter estimation. It characterizes the strength one can borrow across different individuals and impact of data dependence on parameter estimation. Empirically, experiments on both synthetic and real resting state functional magnetic resonance imaging (rs-fMRI) data illustrate the effectiveness of the proposed method.