مقاله انگلیسی رایگان در مورد خودگردان سازی برای مدل های رگرسیون خطی – الزویر 2018

The linear regression model is an important and useful tool in many statistical analyses for studying the relationship among variables. Regression analysis is primarily used for predicting values of the response variable at interesting values of the predictor variables, discovering the predictors that are associated with the response variable, and estimating how changes in the predictor variables affects the response variable (Weisberg, 2005). The standard linear regression methodology assumes that the response variable is a scalar. However, it may be the case that one is interested in investigating multiple response variables simultaneously. One could perform a regression analysis on each response separately in this setting. Such an analysis would fail to detect associations between responses. Regression settings where associations of multiple responses are of interest require a multivariate linear regression model for analysis. Bootstrapping techniques are well understood for the linear regression model with a univariate response (Bickel and Freedman, 1981; Freedman, 1981). In particular, theoretical justification for the residual bootstrap as a way to estimate the variability of the ordinary least squares (OLS) estimator of the regression coefficient vector in this model has been developed (Freedman, 1981). Theoretical extensions of residual bootstrap techniques appropriate for the multivariate linear regression model have not been formally introduced. The existence of such an extension is stated without proof and rather implicitly in subsequent works (Freedman and Peters, 1984; Diaconis and Efron, 1983). In this article we show that the bootstrap procedures in Freedman (1981) provide consistent estimates of the variability of the OLS estimator of the regression coefficient matrix in the multivariate linear regression model. Our proof technique follows similar logic as Freedman (1981). The generality of the bootstrap theory developed in Bickel and Freedman (1981) provide the tools required for our extension to the multivariate linear regression model.