COURSEWORK F71SM Statistical Methods Semester 1, 2020/21 Instructions Marking. This assignment will be marked out of 30 and will carry a 15% weight in the final mark for the course. Submission deadline. The deadline for submission of the assignment is 3.30pm on the 27th of November, 2020. Please make regular back-ups of your work and do not leave finalising it until the last minute as no allowance will be made for computer problems. The mark for coursework submitted late, but within 5 working days of the course- work deadline, will be reduced by 30%. Coursework submitted more than 5 work- ing days after the deadline will not be marked. In a case where a student submits coursework up to five working days late, and the student has valid mitigating cir- cumstances, the mitigating circumstances policy will apply. Students should be advised in such cases to submit a Mitigating Circumstances form for consideration by the Mitigating Circumstances Committee. Form of solution. Your solution to the assignment should take the form of a report. It should be prepared using LaTex, MS Word or other word-processing software. Relevant computer code must be included in the report. Length. Reports should not exceed 4 pages in length. How to submit. Assignments must be submitted online on Vision through Tur- nitin. Online submissions will be subjected to a plagiarism check. Information to be provided. Page 1 of your assignment should include the following information (see final page of handout for example): – the text “Department of Actuarial Mathematics and Statistics”; – your full name; – your HWU registration number (Person ID); – the degree for which you are you registered; 1 – the text ‘Statistical Methods: Assignment’; – the names of other students that you discussed the assignment with; – a signed plagiarism declaration following the wording given at the end of this document (you can either add an electronic signature, or just print your name). Assignments which omit a signed plagiarism declaration will automatically be awarded zero marks. Use of English. Answers that are difficult to follow because of poor use of English might lose marks, so it is important to read through your work carefully before handing it in. Figures and tables. These are usually necessary to present the results of empirical work clearly. If you include figures and tables in your report, the discussion of their contents should be contained in the main text and not in captions. Please ensure that figures are easy to interpret and use clear labelling and legends. Group discussions. You may discuss this project with your classmates. However, you must conduct your own independent analyses and write your report indepen- dently of other students in your class. You will need to form your own conclusions in answering both parts of the assignment. The lecturer naturally expects that you will employ methodology that has been presented or discussed in the course but you are also encouraged to exercise curiosity and show originality. Software problems. You should be able to conduct the empirical part of this question by starting with the code examples we have discussed in the course and then adapting them. It is your own responsibility to get R code to run and to learn how to interpret the results of different functions and packages. Plagiarism and collusion. Failure to reference work that has been obtained from other sources or to copy the words and/or code of another student is plagiarism and if detected, this will be reported to the School’s Discipline Committee. If a student is found guilty of plagiarism, the penalty could involve voiding the course. Students must never give hard or soft copies of their coursework reports or code to another student. Students must always refuse any request from another student for a copy of their report and/or code. Sharing a coursework report and/or code with another student is collusion, and if detected, this will be reported to the School’s Discipline Committee. If found guilty of collusion, the penalty could involve voiding the course. Feedback. We will endeavour to provide feedback (both verbally in sessions and in writing on VISION) within 21 days of submission. Please contact the lecturers for individual feedback on particular questions. 2 Purpose of the assignment The purpose of the assignment is to: demonstrate good knowledge, understanding and application of R related to statis- tical methods; find creative solutions/arguments based on specialist knowledge in Statistics; relate findings in a coherent well-argued text that is professionally presented and also employs material beyond what was presented in class. Assessment criteria Broadly, the assessment criteria are as follows 70% or higher (A grade) Structure structures assignment effectively to facilitate development of argument. Content displays extensive, detailed and secure knowledge and understanding of the subject applies mathematical methods fully accurately to support and develop argument demonstrates clear knowledge and understanding of qualitative and quantitative aspects of the question. Argument engages directly with the question and appreciates wider implications and context presents a clear, coherent and persuasive argument based on correct mathematics displays independence or originality of judgment. Expression uses fluent and accurate prose very good standard of presentation. 60-69% (B grade) Structure structures assignment to facilitate development of argument. 3 Content displays extensive and secure knowledge and understanding of the subject applies mathematical methods mostly accurately to support and develop argument demonstrates sound understanding and knowledge of qualitative and quantitative aspects of the question. Argument engages critically with the question and displays appreciation of the wider implica- tions and context presents and develops ideas logically and persuasively demonstrates some independence of judgment and initiative. Expression uses clear and generally accurate prose good standard of presentation. 50-59% (C grade) Structure broadly structures assignment but organisation of ideas and evidence is sometimes determined by material rather than by the need to develop and support a logical argument. Content displays sound and largely accurate knowledge and understanding of subject applies mathematical methods to support and develop argument but there are issues concerning the accuracy of results and the applicability of ideas demonstrates limited understanding and knowledge of qualitative and quantitative aspects of the question. Argument displays understanding of the questions set but may lack sustained focus and ap- preciation of the wider context states ideas but may not develop them sufficiently or order them in a logical se- quence. 4 Expression prose conveys meaning but lacks sophistication needed to present ideas persuasively expression may be clumsy with narrow vocabulary and spelling or grammar errors adequate standard of presentation. 5 Assignment General information: You may make use of all R functions that have been provided in the computer lab sessions, plus standard R functions and any that you might have developed yourself. However, when you write up your assignment, you should explain what you have done by making reference to the underlying statistics. You may lose marks by making reference in the main text to the functions that you have used in R without further explanation. Your report must be written independently of other students in your class and all pro- gramming work must be your own individual work. The following aspects will be considered for marking: R coding, calculations, suitability of plots; comments, discussion and interpretation of results; clarity of writing and general mathematical exposition; giving detailed answers, with appropriate explanations and good structure. Guidelines: Write up your results in clear English. Use clear mathematical formulae to define any distribution, statistical quantity or property that you are analysing. If these are unclear you will lose marks. Reread everything you write to look for vagueness, ambiguity, and illogical reason- ing. This means taking a step back from the detailed analysis. Define clearly all of the assumptions that you are using. 6 Before you start Use the command set.seed(xxxx) in R, where xxxx are the last four digits of your HWU registration number (Person ID). It is very important that this is done before you start the tasks. If later in your work you want to redo the simulations, you must also run the set.seed(xxxx) command again before you do so. This will ensure that your results are unique and also the same each time you run the simulations. Assignments which fail to include this may be awarded zero marks. Tasks The waiting time (in minutes) between successive emails received on a business account during working hours is believed to follow an exponential distribution with parameter (rate) λ = 0.1. (a) (i) Simulate a sample of n = 40 such waiting times, and obtain suitable summary statistics and a histogram of the values. You should store the simulated values for later use. [4 marks] (ii) We now assume that the true parameter λ of the distribution of waiting times is unknown. Produce plots of the likelihood function and the log-likelihood function, with respect to λ, based on the sample you simulated in part (a)(i). [5 marks] (iii) Determine the maximum likelihood estimate (MLE) of parameter λ in two ways: – by numerically maximising the log-likelihood function in R; – analytically. In both cases, use the sample you simulated in part (a)(i). [4 marks] (iv) We are interested in estimating the probability that the waiting time between successive emails is greater than one hour. Use maximum likelihood estimation to obtain an estimate of this probability using your answers to previous parts. [2 marks] (b) (i) Repeat the simulation in part (a)(i) M = 1000 times, to produce 1000 samples of waiting times, each of size n = 40. For each sample compute and store the value of the MLE of parameter λ. [3 marks] (ii) Obtain suitable summary statistics and a histogram of the distribution of the maximum likelihood estimator of parameter λ, based on your simulation in part (b)(i). [3 marks] 7 (iii) Add the (approximate) theoretical probability density function of the maximum likelihood estimator of λ on the histogram in part (b)(ii), and comment on its distribution. [7 marks] (iv) Comment on how the distribution of the maximum likelihood estimator of λ would change if the simulation involved samples of size n = 100, by also referring to your answer in part b(iii). (You are not required to run this simulation.) [2 marks] [Assignment total: 30 marks] END OF ASSIGNMENT 8 Department of Actuarial Mathematics and Statistics Full name: HWU registration number (Person ID): Degree: F71SM Statistical Methods: Assignment Plagiarism declaration: I confirm that I have read and understood: (a) the note on Plagiarism and collu- sion in the assignment handout; (b) the Heriot-Watt University regulations concerning plagiarism. I confirm that the submitted work is my own and is in my own words. I confirm that any source (aside from course notes and lecture material) from which I obtained information to complete this assignment is listed in the assignment. Any sources not listed in the assignment are listed here: ... Apart from the lecturer, I discussed the assignment and shared ideas with the fol- lowing people: ... Signature Date
欢迎咨询51作业君
