| With the rapid development of economy and innovation of technology,highdimensional data is widely occurring in various fields,such as biomedical science and economics.Recently,a large number of studies have investigated the high-dimensional multi-task regression problems under the assumption of sparsity,that is,exploring the linear relationships among multiple responses and covariates in high-dimensional scenarios.In fields with high reliability requirements,such as clinical research,how to effectively quantify the uncertainty of estimation is of great concern.With the widespread application of bias correction methods in single-response inference problems,some scholars have extended them to multi-response scenarios.However,the existing methods have strict requirements on sample size and sparsity for asymptotic properties analysis.To alleviate the constraints of sample size and the number of nonzero rows of coefficient matrix,we mainly focus on the statistical inference for the unknown coefficient matrix in high-dimensional multi-task regression problems.We establish the definition of the set of important predictors using the concept of distance correlation,and then,based on the hybrid orthogonal ization vectors,we propose a new statistic which is constructed in a row-wise manner.The hybrid orthogonalization vector achieves strict orthogonalization against the column vectors corresponding to the important predictors and relaxed orthogonalization against the column vectors corresponding to the other variables.In this process,by distinguishing important variables from other variables,the proposed method eliminates the influence of important signals,thus enabling the existence of more significant signals.And also,it reduces the requirement on sample size and improves the efficiency of statistical inference.Furthermore,we prove the asymptotic normality of this method and construct confidence intervals for all elements of the unknown coefficient matrix.Numerical experiments show that the new method has significant advantages by combining the performance measures of average coverage probability and average length of confidence intervals.Finally,we apply the proposed method to Alzheimer’s disease data as well as Ovarian cancer data,and the experimental results demonstrate the practical significance in biomedicine,and further validate the effectiveness of the method. |
