Effectiveness Research On Self-born Weighted Least Squares In Robust Linear Regression

Statisticians noted that the probability of appearing gross error is about1%-10%in the production practice and the collected data of scientific experiments. As a mathematical statistical method of processing the relationship among variables, linear regression analysis normally solve the regression coefficients of the regression equation by LS method. Because of the classic LS method uauslly gives the same weight to each observation data, the mishandled of gross errors (abnormal value) produces larger deviation of the regression coefficients, and affects the effectiveness of the regression analysis model. With proposing and developing of the robust estimation theory, many scholars introduced robust estimation theory into the regression analysis model and came up with the concept of robust regression, and pointed out that the robust regression, which has the ability to resist interference of gross errors, can effectively eliminate or weaken the influence of outliers. However, different robust estimation methods have different ability to eliminate or weaken the influence of gross errors on the regression coefficients. Professor Ge presented a new robust estimation method called self-born weighted least squares (SBWLS). With leveling networks and trilateration networks for examples, simulation experiments were conducted. The results indicate that SBWLS can more efficiently eliminate or weaken the effects of gross errors on parameter estimation than other13widely used robust estimation methods.This paper introduced SBWLS into the robust linear regression model, and illustrated its effectiveness. Firstly, the paper take unitary linear regression model and ternary linear regression model as examples, to deduce the computational formula of unitary linear regression model and multiple linear regression model for SBWLS according to the basic principles of SBWLS. Secondly, when observations do not contain gross errors(ε=0.0σ0), the paper employed simulation experiments, taking unitary linear regression and multiple linear regression (binary to five linear regression) with different numbers of observations as examples, to illustrate the accuracy loss of SBWLS and other13commonly used robust estimation methods by calculating and comparing the average relative gain of SBWLS and other13commonly used robust estimation methods relative to LS. Then, when observations contain gross errors, the paper employed simulation experiments, taking unitary linear regression and multiple linear regression (binary to five linear regression) with different numbers of observations, different numbers of gross errors (1-3) and different values of gross errors (5.0σ0and10.0σ0) as examples, to compare the ability of SBWLS and other13kinds of commonly used robust estimation methods to eliminate or weaken the impact of gross errors on regression coefficient by calculating the average relative gain of SBWLS relative to LS and other13commonly used robust estimation methods, and illustrate the validity of SBWLS used in unitary linear regression model and multiple linear regression model (binary to five linear regression).The results indicate that SBWLS can more efficiently eliminate or weaken the influences of gross errors on regression coefficient than other13commonly used robust estimation methods; SBWLS is a relatively more efficient method for robust linear regression model.