1. Gu Yuxian Wang Weinan Beijing National Day School

2. Part 1 The Simple Linear Regression Given two variables X and Y. ， … are measured without an error, … are measured with error So we can let We can use the least squares estimators and the maximum likelihood estimator to estimate parameter and.

3. The Least Squares Estimators Let All we need to do is to minimize Δ. Let, Solve the equation.

4. The Maximum Likelihood Estimator Assume that So

5. The likelihood function Compute and Solve We get

6. Efficiency Analysis They are unbiased.

7. Part2 Errors-in-Variables (EIV) Regression Model When the measurements for X is not accurate. There are two ways to measure errors. The orthogonal regression and the geometric mean regression.

8. The Orthogonal Regression(OR) The distances between the regression line and points are To minimize Compute and solve We are supposed to get

9. The Geometric Mean Regression(GMR) The area is To minimize Compute and solve we get

10. Parametric Method Assume X and Y follow a bivariate normal distribution We use moment generating function (mgf) to derive the distribution of X and Y :

11. Since are independent, we can separate mgf. The bivariate normal distribution that method of moment estimator(MOME)

13. We get:

14. Special Situation for MLE The Orthogonal Regression(OR) The Geometric Mean Regression (GMR)

15. – This is when Y has no error. – This is when X has no error, so we get the same answer as our first discussion.

16. Another Estimator We want to (1)occupy all la (like MLE) (2)without distributions(like (OR)&(G)) Calculate

17. Let So is increasing and We get Prove 1-1 to

18. Let So there is at least one root for Prove 1-1 to We have

19. So there is ONLY one root for (when ) And when Then we have We can proof

20. Another Estimator Again The angle Let Compute & solve We get ***

21. Part3 Multiple Linear Regression The Least Squares Estimators Similar to simple linear regression: Compute We will get a group of equations:

22. Assume its coefficient matrix is The solution is

23. Errors-in-Variables (EIV) Regression Model(Two Variables) The Orthogonal Regression(OR) The Geometric Mean Regression(GMR1)(the volume ) The Geometric Mean Regression(GMR2)(the sum of area )