1. Course content Course Description Statistical Methods, ST741A, 7.5 hp Department of Statistics Autumn, 2017 This course introduces several statistical techniques that might be used as bits of methodological part for writing M.Sc. thesis. Course content and instructors vary from year to year. Reading material is distributed at Mondo by instructors before their corresponding part starts. Students are expected to write 1-3 (depending on the number of teachers involved in the course but at least one report per teacher) home assignments 5-15 pages long to demonstrate understanding of methods presented. 2. Intended learning outcomes After the course, the students should be able to: Write 5-15 page essays based on statistical methodology introduced during lessons; the level of writing should be at the level suitable to M.Sc. thesis writing Follow selected research papers and reproduce presented research results 3. Literature and plan for the lectures Research papers and other material provided by instructors; Attend introductory lecture is strongly advisable Selected papers to be used during the course: G.Casella, E.I.George (1992) Explaining the Gibbs Sampler, The American Statistician, vol. 46, No. 3, pp. 167-174. 1
A.F.M. Smith and G.O. Roberts (1993) Bayesian Computation via the Gibbs Sampler and related Markov Chain Monte Carlo Methods, Journal of the Royal Statistical Society, Series B (Methodological), vol. 55, no 1, pp 3-23. AA part description: MCMC-type simulation techniques (October 3, 23, 24, 26) Bayesian computations via Gibbs Sampler and related Markov Chain Monte Carlo methods are in the core of this part of the course. The use of the Gibbs sampler for Bayesian computations is reviewed, explained and illustrated. Some other Markov Chain Monte Carlo simulation methods are also briefly described. Several techniques to check convergence of suggested chains are mentioned. We will discuss number of examples for conjugate Bayesian analysis. Special attention will be devoted to understanding Gaussian case: majority of material in this section will be left for home reading. Student will be provided with several research papers that will be expected to be read in detail. Upon completion of this part of the course, student will deepen its knowledge in Bayesian statistical paradigm and should be able to read some published research papers in the area and write own program to implement simple Gibbs sampler. A home assignment will be provided upon completion of all four lectures and students will get up to 10 days to complete the task. The max amount of points from this part is 34p and it is 17p min one needs to get in order to pass. FM part description: Advanced methods for sample size determination (Oct. 5, 6, 10) Determining sample size for an observational study or an experiment is an important statistical method because samples that are too large may waste time, resources, and money, while samples that are too small may lead to inaccurate results. Based on ensuring a certain power for a significance test (according to the Neyman-Pearson philosophy), we recall the statistical principles and formulae for sample size determination. Then we focus on problems occurring in applications when determining sample size. Sample size formulae depend on parameters which are usually not known before data is collected. Therefore, the statistical problem is often not resolved with knowing the sample size formula the challenge for the 2
statistician in practice is to apply the formula in a given context in a meaningful and robust way. We introduce advanced methods for sample size determination. One method deals with the uncertainty around the parameters by assuming a prior distribution for them. A further method we will discuss is a decision theoretic approach to determine an optimal sample size. We will introduce in the lectures the decision theoretic background which we need here. The sample size determined by the decision theoretic approach depends on assumed parameters as well and we will discuss how we deal with this in real life applications. A home assignment for this topic will be provided after the three lectures on Oct 10. Deadline to hand in will be Oct 16. The maximum number of points which can be obtained is 33 and 16.5 points are required to pass this part. HN part description: Asymptotics (Oct. 13, 17, and 19) This is a finite sequence of lectures about infinite sequences of random variables. Complicated probability distributions often appear when building probability models and when analysing data. The methods provided at these lectures give us some tools for approximating probability distributions and can thus be useful in model building and data analysis. The theme of the first lecture is What is this thing called convergence?. We will take a walk along a coordinate axis and see how distribution functions changes as we proceed. At this excursion, we will also find things like consistency and sequences of expectations that converge to something unexpected. The theme of the second lecture is Where does the normal distribution come from, and where does it go?. The normal distribution is perhaps the most used probability model in statistics. At this lecture, we will look at the arguments, based on limit theorems, which motivate using normal distributions. We will also see that similar arguments may lead to other limit theorems and to distributions that are highly non-normal. The theme of the third lecture is Going extreme. The central limit theorems concern the asymptotic distributions of averages of random variables. However, other statistics than averages may be of interest when constructing probability models and analyze data. In this lecture, we consider limit distributions of maxima and minima of sequences of random variables. 3
To each lecture will be a homework assignment that covers some theory of the lectures topic. Solutions are supposed to be presented at the next lecture. The max amount of points from each assignment is 11. A total of at least 16.5 points from these assignments is required in order to pass. Table 1 is a preliminary and a tentative plan for the teaching schedule. Lecturer reserves the right to make appropriate adjustments during the course. Table 1: Preliminary plan for teaching Lecture Date Content/teacher Time/B705 F1 3October AA+HN+FM: Introduction important!! 15-17 F2 05Oct FM: 13-16 F3 06Oct FM: 13-16 F4 10Oct FM: 8-10,12-13 F5 13Oct HN: 9-12 F6 17Oct HN: 9-12 F7 19Oct HN: 9-12 F8 23Oct AA: 14-17 F9 24Oct AA: 14-17 F10 26Oct AA (summary and home 14-17 assignment): F11 F12 F13 4. Examination and criteria for assessment Exam1: three written assignments Exam2: students can complement above assignments or will be given additional tasks by instructor(s) Students are assessed with three home assignments that are distributed by each instructor upon completion of lectures, see schedule. Maximum number of points is 100 points. Students are expected to get at least half of the points from each of the three home assignments, i.e. getting best grade from two parts of the course and failing third part will result in overall fail mark. 1) Part III (AA): 17 points minimum out of 34 to pass this part 2) Part I (FM): out of 33 points 3) Part II (HN): out of 33 points 4
What to expect at the written exam: The home assignment is an individual assessment: collaboration is allowed but the written report is an individual effort. Note that your submission may be tested by text matching software of TurnitIn-type to discover plagiarism. If plagiarism is established, the evaluation is set to be 0 zero, final grade Fail and no second chance given during the same term. To pass the entire course, the student must collect at least 17 points from AA part and 16.5 points from parts FM and HN. In other words, the student will not pass the course if he/she fails any of the three parts. Credits for home assignment cannot be moved to the next term. The grade is based on the total score. The following seven criteria-referenced grades are used: A B C D E F Excellent Very Good Good Satisfying Sufficient Insufficient A (Excellent): The student should be in a proper and well-structured way to apply statistical methods and associated statistical inference that are not necessarily directly addressed in the course material. The student is also clearly able to present and interpret his/her results; explain concepts, methods and theories used in the implementation of these methods. B (Very good): The student will correctly and in a well-structured way be able to apply the statistical methods and associated statistical inference that is directly addressed in the course material. The student is also clearly able to present and interpret his/her findings; explain the concepts, methods and theories used in the implementation of these methods. C (Good): The student will correctly and in a well-structured way be able to apply the statistical methods and associated statistical inference that are directly addressed in the course material. The student should also be in a good way to present and interpret his/her findings; explain concepts, methods and theories used in the implementation of these techniques. D (Satisfying): The student will be able to apply statistical methods with related statistical inference that are directly addressed in the course material. The student will forward in a satisfactory way to present and interpret his/her findings; explain concepts, methods and theories used in the implementation of these techniques. E (Sufficient): The student will be able to apply statistical practices directly addressed in the course material. The student, in a satisfactory way, will present and interpret his/her findings; explain the concepts, methods and theory used in the implementation of these techniques. F (Insufficient): The student cannot correctly apply statistical methods that have been considered in the course. A single final grade for the complete course will be given according to Table 3. 5
Table 3: The sum of the points from three home assignments and the final grade Points Grade 91-100 A 80-89 B 70-79 C 60-69 D 50-59 E 0-49 F Lectures and Examiners: Andriy Andreev: Andriy.Andreev@stat.su.se Hans Nyquist: Hans.Nyquist@stat.su.se Frank Miller: Frank.Miller@stat.su.se 6