The Analysis of Bootstrap Method in Linear Regression Effect

Abstract

This paper combines the least squaress estimate, least absolute deviation estimate, least median estimate with Bootstrap method. When the overall error distribution is unknown or it is not the normal distribution, we estimate the regression coefficient and confidence interval of coefficient, and through data simulation, obtain Bootstrap method, which can improve stability of regression coefficient and reduce the length of confidence interval. This paper focuses on linear regression. The traditional regression analysis method assumes that the regression equation is y i = β 0 + β 1 x i + ε i , (i = 1, 2, · · · , n), where random error ε i ∼ N(0, σ 2). Under the assumption of error normality, the coefficient β can be estimated, while in the significant test of regression, the corresponding distribution of test statistics is obtained. However, when the error ε is not normal or its distribution is unknown, how to estimate the regression coefficient ,how to estimate the confidence interval of coefficient and how to significantly test the regression equation? The following uses Bootstrap method to solve the above problems. According to the different regression relationships, Bootstrap re-sampling methods can be divided into two types. 1.1 Model-based Bootstrap regression The independent variable x in correlation model regression is a controllable variable(general variable) and only y is a random variable. Random sampling error is ε i , (i = 1, 2, · · · , n). ε i in regression equation accords with Gauss-Markov assumption: E(ε i) = 0; Var(ε i) = σ 2 ; Cov(ε i , ε j) = 0, (i j) (1) But ε i is not always a normal distribution. It is be noted that σ 2 is not the variance of residual e i = y i − ˆ y. Normalize the residual e i to obtain the revised residual r i = e i − E(e i)/ √ Var(e i), (i = 1, 2, · · · , n). In order to better model the actual distribution of residual with the experience distribution, the revised residual can be centralized. Denoted the revised residual after centralizing r = n i=1 r i by r i = r i − r.