Dimension reduction of high dimensional data plays a significant role in many application fields, and it is a current research focus in statistics. This paper studies dimension reduction of high dimensional data from three aspects: heterogeneous data, matrix data, and discretization of continuous variables. Heavy tailed distribution, outliers or influence points are often encountered in the study of high-dimensional data, and the general regression model is not a simple linear relationship between the response variables and covariates. Quantile regression models can deal with these problems well. Due to the rapid development of data collection technologies, types of data collected are becoming more and more diverse. Matrix valued data has been widely used in many fields in recent years, such as EEG data, brain functional magnetic resonance imaging data. Therefore, matrix-valued data has become one of the hotspots in statistical research. The row and column of matrix data usually have special meanings, but few literatures have studied the row and column effect of matrix data.This paper studies the inference of row and column effects of high-dimensional matrix-valued data. Furthermore, discretization is a useful method for dimension reduction of high-dimensional data. But for the deviation caused by discretization, there are few researches of the existing literatures. We make an in-depth study of this problem in this paper.

Firstly, in high dimensional quantile regression, it is commonly assumed in the literature that the parameter βτ ∈ Rp is sparse and that predictors are weakly correlated, where τ ∈ (0, 1) is a given level. The sparsity assumption, however, is hard to verify in practice and can be violated, and the predictors can be highly correlated in many applications, so that the eigenvalues of the covariance matrix decrease rapidly, or they are approximately sparse. Different from the existing estimators in the literature that estimate all entries of βτ simultaneously, we in this paper consider the problem of estimating a single element or a small part of βτ, proposing a new estimator called partial quantile regression (PQR). Interestingly, it is found that the approximately sparse of the eigenvalues can be beneficial in our framework, relaxing greatly the sparsity assumption on βτ. The statistical properties of the estimator are established. Simulation results and real data analysis confirm the effectiveness of the estimator.