Below will be given exercises for the coming week(s) and an overview of exercises that have been given to earlier weeks. Unless otherwise stated, exercises are from the book by Casella and Berger.
Exercises to coming week(s)
Week 48 (30 November):
This is the last week of teaching, and we will consider the following exercises:.
- Exam autumn 2013: Problems 1 and 2.
- Exam autumn 2014: Problems 1 and 2.
Exercises to earlier weeks
Week 47 (Solutions):
- Do the following exercises from chapter 9: 9.4 and 9.6.
- Additional exercise: Let be independent and Poisson distributed with parameter \(\lambda\). Derive approximate \(1-\alpha\) confidence intervals for \(\lambda\) by inverting
- a) the Wald test
- b) the score test
- c) the likelihood ratio test
- Exam autumn 2015: Problems 1, 2, and 4 (solutions)
- Do the following exercises from chapter 8: 8.15, 8.22ac, 8.25bc, 8.31 and 8.34b.
- Do the following exercise from chapter 10: 10.34a.
- Additional exercise: Look at slide 21 from the lectures in week 45. Derive an expression for \(-2\log \lambda({\bf X})\).
- Do the following exercises from chapter 8: 8.3, 8.6 and 8.37.c. (Before you do exercise 8.37.c, you should read example 8.2.6.)
- Do the following exercises from chapter 10: 10.1, 10.5, 10.6 and 10.9.
- Do the following exercises from chapter 5: 5.33, 5.35, 5.41, 5.42 and 5.44.
- Do the following exercises from chapter 7: 7.47, 7.52, 7.59 and 7.60.
- Additional exercise: Assume that \(X_1, X_2, \ldots , X_n\) are independent and Poisson distributed with mean \(\lambda\). a) Find a sufficient and complete statistic for \(\lambda\). b) Find an unbiased estimator for \(\tau(\lambda)=\exp(-\lambda)\) based on \(X_1\). c) Find the best unbiased estimator for \(\tau(\lambda)\).
- Do the following exercises from chapter 7: 7.19, 7.38, 7.41 and 7.42.
- Do the following exercises from chapter 7: 7.9, 7.11, 7.13, 7.22, 7.24 and 7.40. (In order to find the mean and the variance of the MLE in exercise 7.11, you should note that the MLE is of the form n/T, where T is gamma-distributed with shape parameter n and scale parameter 1/theta, and then use the result of exercise 3.17.)
- Do the following exercises from chapter 6: 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8 and 6.9a-c.
- Do the following exercises from chapter 5: 5.5, 5.6b, 5.11, 5.13, 5.17, 5.18a-b, 5.24, 5.25. [Before you do exercise 5.11 you should take a look at Jensen's inequality (page 190). In exercise 5.13 you may use that \((n-1)S^2/\sigma^2 \) is chi-squared distributed with n-1 degrees of freedom together with the result in exercise 3.17 (which is valid for \(\nu >-\alpha\)) and the fact that the chi-squared distribution is a special case of the gamma distribution. Before you do exercise 5.17 you should read section 5.3.2 (pages 222-225). In 5.17a you should start with U and V independent and chi-squared distributed variables with p and q degrees of freedom, and consider the transformation X=(U/p)/(V/q) and Y=U+V. In 5.17b you may again use the result in exercise 3.17.]
- Do the following exercises from chapter 4: 4.21, 4.22 and 4.27.
- Additional exercise: Look at slide 5 from the lectures on multivariate distributions. a) Prove the result on moment generating functions stated on the first half of the slide. b) Prove the result on the distribution of linear combinations of independent normal random variables given at the bottom half of the slide.
- Additional exercise: Prove formula (3.3.17) in Casella and Berger. Hint: Write \(\Gamma(\alpha)\Gamma(\beta)\) as a double integral and show by a suitable change of variables that the double integral equals \(\Gamma(\alpha+\beta)B(\alpha,\beta)\).
- Exercises on exponential family of distributions: 3.28b-d (consider only the case where both parameters are unknown in b and c), 3.30b, and 3.33b-c. In exercise 3.30b one should replace "beta(a,b)" by "Poisson(\(\lambda\))", see the errata to Casella and Berger.
- Do the following exercises from chapter 4: 4.4a-b and 4.5.
Week 35 (Solutions):
- ???Read section 2.1 and do exercises 2.1 and 2.6.a.
- Read pages 62-68 in section 2.3 and do exercises 2.30, and 2.33.
- Do the following exercises from chapter 3: 3.13, 3.17, 3.38, and 3.39.