The Construction And Comparisons Of Simultaneous Confidence Tubes For Multivariate Linear Regression

Simultaneous confidence bands are a direct generalization 0f using confidence intervals to bound an unknown parameter and the differences between unknown parameters,and have been extensively studied by many experts and scholars and well developed a useful statistical inferential tool that can be used in many statistical branches.While the papers in simultaneous bands are numerous,most of them focus on multiple linear regression with only one response variable in linear regression models,little people study multivariate linear regression models having several response variables.However,when studying practical problems,we could encounter complex situations that have several response variables sometimes.So,it is necessary to study problems with several response variables.This paper summarizes the algorithm of exact simultaneous confidence bands for multivariate linear regression proposed by Liu et al.in 2016,which based on the theories of simultaneous confidence bands for multiple linear regression.This paper also computes the critical constant by simulation method and gives an example to observe the shape and position of simultaneous confidence band.Because of the shape of the band,tube is more pertinent to call it.The expression by simplifying simultaneous confidence tube shows that its value has nothing to do with regression coefficients and error covariance but relates to the numbers of independent variable and observe data,also confidence level and simulation numbers,when the covariate region is free.Comparing the actual frequency that simultaneous confidence tube covers the true linear regression line with the given confidence level and varying the values of related factors to carry out experimental design,we have the following conclusion.For the multivariate linear regressions having only one independent variable and two response variables over the free covariate range,while the numbers of observed data and given confidence level have a strong influence on the accuracy of simultaneous confidence tube,the influence of simulation numbers computing the covering frequency is smaller.