Complex Data Constraints The Linear Model Of Statistical Inference

In this paper, we first generalize the development and main property of linear regression model and introduce statistical inference for the linear regression model whose regression coefficients with constraints. Based on these theories, we study the measurement error model (error in independent variable and the dependent variable can be accurately observed) and variable with missing values model (missing values in dependent variable and independent variable can be completely observed). Then we introduce the development of these two models, give the parameter estimation for these two complex data models, and derive the constrained least squares estimation for the parameters to be estimated with constraints. By Lagrange multiplier method, we construct the test statistic based on the difference of residual sum of squares between null hypothesis and alternative hypothesis of constraints and Wald statistic to test the constraints, we conclude that the test statistic we have constructed follow the chi-square distribution when null hypothesis established, computer simulations are used to examine the performance of our procedure and the results are satisfactory. At last, we summarize the work in this paper and outlook the future development of regression model.