1. Q Is a P P Orthogonal Matrix If Qx = X . Prove That Q Q=QQ =I

Sample of Test Questions:

1. Q is a p×p orthogonal matrix if |Qx|=|x|. Prove that Q’Q=QQ’=I.

2. Define Householder transformations. What is it used for?

3. Show that if H is a composition of Householder transformations then H is orthogonal.

4. Let X be a n×p data matrix with n observations and p variables. Define the QR decomposition.

5. How do you use Houlsholder transformations to calculate the QR decomposition of a data matrix X.

6. How do you apply Given’s rotations to calculate the QR decomposition of a data matrix X.

7. Show how to compute the Least Squares regression estimate using the QR decomposition.

8. Define the singular value decomposition of a matrix M.

9. Let X be a n×p data matrix with n observations and p variables. Show how to use the SVD to calculate the principal components of the data. 10. What do you need to assume about n and p in prob. 9? Suppose that n < p, can you figure out how to calculate the principal components of X.

11. Suppose that A is a nonsingular covariance matrix. Show that the sequence

xi+1= Axi/|xi| converges to the largest eigenvalue of A.

12. Write the logistic regression model. What is the model with link function? What is the variance function?

13. Which algorithm should be used to calculate the MLE estimator of the logistic model?

14. What is the difference between fix effects and random effects?

15. What is a mix effects model? What do we estimate?

16. Define the 3 and 4 parameters Emax model. Find a relationship to the logistic model.

17. Draw a picture of the Emax dose response model and show the parameters on the graph.