Smoothed tensor quantile regression estimation for longitudinal data

As extensions of vector and matrix data with ultrahigh dimensionality and complex struc-tures, tensor data are fast emerging in a large variety of scientific applications. In this paper, a two-stage estimation procedure for linear tensor quantile regression (QR) with lon-gitudinal data is proposed. In the first stage, we account for within-subject correlations by using the generalized estimating equations and then impose a low-rank assumption on ten-sor coefficients to reduce the number of parameters by a canonical polyadic decomposition. To avoid the asymptotic analysis and computation problems caused by the non-smooth QR score function, kernel smoothing method is applied in the second stage to construct the smoothed tensor QR estimator. When the number of rank is given, a block-relaxation al-gorithm is proposed to estimate the regression coefficients. A modified BIC is applied to estimate the number of rank in practice and show the rank selection consistency. Further, a regularized estimator and its algorithm are investigated for better interpretation and ef-ficiency. The asymptotic properties of the proposed estimators are established. Simulation studies and a real example on Beijing Air Quality data set are provided to show the per-formance of the proposed estimators.(c) 2022 Elsevier B.V. All rights reserved.