Multivariate factorisable sparse asymmetric least squares regression

More and more data are observed in form of curves. Numerous applications in finance, neuroeconomics, demographics and also weather and climate analysis make it necessary to extract common patterns and prompt joint modelling of individual curve variation. Focus of such joint variation analysis has been on fluctuations around a mean curve, a statistical task that can be solved via functional PCA. In a variety of questions concerning the above applications one is more interested in the tail asking therefore for tail event curves (TEC) studies. With increasing dimension of curves and complexity of the covariates though one faces numerical problems and has to look into sparsity related issues. Here the idea of FActorisable Sparse Tail Event Curves (FASTEC) via multivariate asymmetric least squares regression (expectile regression) in a high-dimensional framework is proposed. Expectile regression captures the tail moments globally and the smooth loss function improves the convergence rate in the iterative estimation algorithm compared with quantile regression. The necessary penalization is done via the nuclear norm. Finite sample oracle properties of the estimator associated with asymmetric squared error loss and nuclear norm regularizer are studied formally in this paper. As an empirical illustration, the FASTEC technique is applied on fMRI data to see if individual's risk perception can be recovered by brain activities. Results show that factor loadings over different tail levels can be employed to predict individual's risk attitudes.