Practice Problems

Extra problems

17 Jan: Extra Problems Set #1
31 Jan: Extra Problems Set #2
28 Feb: Extra Problems Set #3
25 Mar: Extra Problems Set #4

24 Apr: (To be updated)

  • 1. 8.13 (parts a - c). Also find the distributions of SSA and SSB(A).
  • 2. Consider the Aitken model whose mean and covariance forms are given in the last equation on page 200 in the book. Here x* is the new covariate and we want to predict y*. Assuming the data is generated from a normal distribution, find the BLUP using the conditional distribution approach.
  • 3. Consider the mixed effects model y = Xb + Zu + e, where u ~ N(0, G), e ~ N(0, R), u and e are independent. Find the joint distribution of Y and u. Using this, find BLUP of u using the conditional expectation argument. Also, what is the GLS estimator of b?
  • 4. Consider the intercept only Gauss-Markov model (i.e., X only has one column of 1s). Find the BLUP of a new observation y* and its prediction interval. As sample size n increases, what happens to the lower and upper limits of the prediction interval? If you had an infinite sample size, can you make an exact prediction with zero MSPE?
  • 5. 8.9 (just write the SS terms and compute their distributions). Can you still separate SSC and SSE in this case?

 

Problems for lab sessions

Date Problems
 22nd Apr 1. Consider the model given in 8.3, and the corresponding ANOVA table. Find the distribution of SSA. [Hint: start with the joint distribution of appropriate group means]

2. 8.10
15th Apr
 1. Find the expected squared length of the simultaneous Bonferroni intervals for one way ANOVA with a=3 groups and b=5 units per group.

2. 6.14 (a) and (b)

3. 8.6
8th Apr 1. We will discuss Exam 2 problems and any specific questions you have.

2. No new lab problems will be discussed (since we have not completed LRT and F-test yet).
25th Mar
 Lab problems
18th Mar
 Lab problems
11th Mar
 No lab
4th Mar 1. Go over Lagrange Multiplier (Appendix B; just the results)

2. Problem 2.2 and Example B.3 (using Lagrange Multiplier)

3. Using Lagrange multiplier method, minimize the least squares criterion Q(b) with respect to the constraint Cb = 0.

4. 4.25
25th Feb No lab
18th Feb No lab
11th Feb Midterm #1
4th Feb
 Lab problems
28th Jan
 1. A.56

2. A.56

3. A.58

4. A.59

5. A.60

6. A.49

7. 2.10 (using A.37)
14th Jan 1. Show that C(A) is the orthogonal complement of N(A^T)

2. A.16

3. A.36

4. A.37

5. A.39

6. A.53

7. A.54