The University of Nottingham SCHOOL OF MATHEMATICAL SCIENCES
A LEVEL 4 MODULE, SPRING SEMESTER 2019-2020 COMPUTATIONAL STATISTICS
Suggested time to complete: TWO Hours THIRTY Minutes
Answer ALL questions
Your solutions should be written on white paper using dark ink (not pencil), on a tablet, or typeset. Do not write close to the margins. Your solutions should include complete explanations and all intermediate derivations. Your solutions should be based on the material covered in the module and its prerequisites only. Any notation used should be consistent with that in the Lecture Notes.
Guidance on the Alternative Assessment Arrangements can be found on the Faculty of Science Moodle page: https://moodle.nottingham.ac.uk/course/view.php?id=99154#section-2
Submit your answers as a single PDF with each page in the correct orientation, to the appropriate dropbox on the module’s Moodle page. Use the standard naming convention for your document: [StudentID]_[ModuleCode].pdf. Please check the box indicated on Moodle to confirm that you have read and understood the statement on academic integrity: https://moodle.nottingham.ac.uk/pluginfile.php/6288943/mod_ tabbedcontent/tabcontent/8496/FoS%20Statement%20on%20Academic%20Integrity.pdf
A scan of handwritten notes is completely acceptable. Make sure your PDF is easily readable and does not require magnification. Text which is not in focus or is not legible for any other reason will be ignored. If your scan is larger than 20Mb, please see if it can easily be reduced in size (e.g. scan in black & white, use a lower dpi — but not so low that readability is compromised).
Staff are not permitted to answer assessment or teaching queries during the assessment period. If you spot what you think may be an error on the exam paper, note this in your submission but answer the question as written. Where necessary, minor clarifications or general guidance may be posted on Moodle for all students to access.
Students with approved accommodations are permitted an extension of 3 days.
The standard University of Nottingham penalty of 5% deduction per working day will apply to any late submission.
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Academic Integrity in Alternative Assessments
The alternative assessment tasks for summer 2020 are to replace exams that would have assessed your individual performance. You will work remotely on your alternative assessment tasks and they will all be undertaken in “open book” conditions. Work submitted for assessment should be entirely your own work. You must not collude with others or employ the services of others to work on your assessment. As with all assessments, you also need to avoid plagiarism. Plagiarism, collusion and false authorship are all examples of academic misconduct. They are defined in the University Academic Misconduct Policy at: https://www.nottingham.ac. uk/academicservices/qualitymanual/assessmentandawards/academic-misconduct.aspx
Plagiarism: representing another person’s work or ideas as your own. You could do this by failing to correctly acknowledge others’ ideas and work as sources of information in an assignment or neglecting to use quotation marks. This also applies to the use of graphical material, calculations etc. in that plagiarism is not limited to text-based sources. There is further guidance about avoiding plagiarism on the University of Nottingham website.
False Authorship: where you are not the author of the work you submit. This may include submitting the work of another student or submitting work that has been produced (in whole or in part) by a third party such as through an essay mill website. As it is the authorship of an assignment that is contested, there is no requirement to prove that the assignment has been purchased for this to be classed as false authorship.
Collusion: cooperation in order to gain an unpermitted advantage. This may occur where you have consciously collaborated on a piece of work, in part or whole, and passed it off as your own individual effort or where you authorise another student to use your work, in part or whole, and to submit it as their own. Note that working with one or more other students to plan your assignment would be classed as collusion, even if you go on to complete your assignment independently after this preparatory work. Allowing someone else to copy your work and submit it as their own is also a form of collusion.
Statement of Academic Integrity
By submitting a piece of work for assessment you are agreeing to the following statements:
1. I confirm that I have read and understood the definitions of plagiarism, false authorship and collusion.
2. I confirm that this assessment is my own work and is not copied from any other person’s work (published or unpublished).
3. I confirm that I have not worked with others to complete this work.
4. I understand that plagiarism, false authorship, and collusion are academic offences and I may be referred to the Academic Misconduct Committee if plagiarism, false authorship or collusion is suspected.
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1. (a)
1 MATH4007-E1 i) The truncated Poisson dsitribution has probability mass function
𝑝(𝑥;𝜆) = 𝜆𝑥𝑒−𝜆 , 𝑥 = 1,2,3,…, 𝑥!(1 − 𝑒−𝜆)
where 𝜆 > 0 is a parameter. Prior information about 𝜆 is summarized by 𝑝(𝜆) ∝ 𝑒−|𝜆−3|, 𝜆 > 0.
Given observed data 𝑥1 = 2, 𝑥2 = 5, 𝑥3 = 13, derive the log posterior distribution,
denoted by 𝑙(𝜆), up to an additive constant.
ii) It is required to find the maximum of 𝑙(𝜆). An initial interval thought to contain a
maximum is given by 𝜆1 = 4.36, 𝜆3 = 5.81. Carry out two iterations of the Golden
Ratio method, i.e. find the next two intervals containing a maximum of 𝑙(𝜆). iii) What is the statistical interpretation of the output of the algorithm?
(b) The joint density of two random variables 𝑋 and 𝑌 is given by 𝑓(𝑥,𝑦)∝𝑥2𝑦2exp(−𝑥𝑦−5𝑥−4𝑦), 𝑥,𝑦>0.
i) Derive the Laplace approximation to the marginal density 𝑓 (𝑥).
ii) Evaluate this for the case 𝑥 = 2.
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iii) Treating 𝑦 as missing information, give full details of how the EM algorithm can be used to find the mode of the marginal distribution 𝑓 (𝑥).
iv) Starting from an initial value 𝑥(0) = 0.1, perform two iterations of the EM algorithm. [25 marks]
[15 marks]
2 MATH4007-E1 2. (a)Itisrequiredtosamplefromadensity𝑓,where
𝑓(𝑥)= (1−𝑥), 0<𝑥<1, 𝑐
and 𝑐 > 0 is a constant. i) Find the constant 𝑐.
ii) Hence, explain how to sample from 𝑓 using inversion.
iii) Produce one sample from 𝑓, given a sample 𝑈 = 0.4 from a 𝑈 (0, 1) distribution.
(b) The joint density of two random variables 𝑋 and 𝑌 is given by 𝑓(𝑥,𝑦)∝𝑥2𝑦2exp(−𝑥𝑦−5𝑥−4𝑦), 𝑥,𝑦>0.
i)
ii)
iii)
(c) i)
ii)
Show that the marginal density of 𝑋 is proportional to 𝑥2𝑒−5𝑥 .
(𝑥+4)3
The density of a random variable 𝑍 which follows a Gamma distribution with parameters
[8 marks]
𝑎 and 𝑏 is
𝑝(𝑧) ∝ 𝑧𝑎−1 exp{−𝑏𝑧}.
Show how samples from the marginal distribution of 𝑋 can be obtained using the rejection algorithm, using samples from a Gamma distribution with 𝑎 = 3, 𝑏 = 5. Assume that samples from the conditional distribution 𝑝𝑌 |𝑋(𝑦|𝑥) can be obtained for any value of 𝑥. (You do not have to find this distribution or how to sample from it.) Explain how this and the above result can be used to sample from the joint density of 𝑋 and 𝑌 to estimate 𝐸[𝑋𝑌 ].
[20 marks]
Consider the 𝑘 Nearest Neighbour estimator of a density 𝑓. Carefully describe the rationale behind this estimator, and discuss the influence of 𝑘 on the resulting estimates.
Three data points from an unknown density 𝑓 are observed, 𝑥1 = 1, 𝑥2 = 2 and 𝑥3 = 4. Calculate the 𝑘 Nearest Neighbour estimator of 𝑓, with 𝑘 = 2.
[12 marks]
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3. (a) Consider the density
𝑓(𝜃,𝜙)∝𝜃2𝜙2exp{−𝜃𝜙−5𝜃−4𝜙}, 𝜃>0,𝜙>0.
i) Find the full conditional distributions 𝑓 (𝜃|𝜙) and 𝑓 (𝜙|𝜃).
ii) Hence, describe a Gibbs sampler to sample from 𝑓.
iii) Suppose instead that the Metropolis-Hastings algorithm is to be used to sample from 𝑓. Describe fully a random walk Metropolis algorithm which updates both variables simultaneously, using proposals of the form
𝜃′ ∼𝑁 𝜃,𝑆. (𝜙′) 2((𝜙))
As part of your answer, discuss the role of 𝑆 on the performance of the sampler. iv) Suppose 𝑆 = 𝐼2, the identity matrix, and the chain is currently in the state 𝜃 = 2,
𝜙 = 1. Given 3 independent 𝑈 (0, 1) random numbers 𝑍 =0.2, 𝑈 =0.8, 𝑉 =0.45,
perform one update of the chain described in the previous part.
(b) The following data are available, which are believed to be random samples from a
population with mean 𝜇 = 7
9.1 5.8 5.1 9.7 5.5 4.3 6.0
i) Explain why a randomisation test might be preferred to a t-test in order to test the hypothesis 𝐻0 ∶ 𝜇 = 7.
ii) Describe a suitable randomisation test to test 𝐻0 ∶ 𝜇 = 7, stating any assumptions you make.
iii) Use the 𝑈 (0, 1) random numbers
{0.46, 0.84, 0.02, 0.76, 0.67, 0.53, 0.22}
in order to calculate one replicate of your test statistic for the test in (ii).
[12 marks]
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[28 marks]