MATH4007 COMPUTATIONAL STATISTICS
Assessed Coursework 1 — 2022/2023
Your work should be submitted electronically via the module’s Moodle page by 15:00 Thursday
15th December 2022. Since this work is assessed, your submission must be entirely your own
work (see the University’s policy on Academic Misconduct). Submissions up to five working days
late will be subject to a penalty of 5% of the maximum mark per working day.
Submission requirements
The submission should be uploaded electronically via the submission box on Moodle, and contain:
1. A pdf file containing any computational results (plots/relevant output) and discussion. This
can be produced using e.g. R Markdown, or by copying output into a Word document.
Please convert any documents to pdf for uploading.
2. A pdf of your theoretical working. A scan of handwritten work is fine, but you could also
typeset using Latex if you prefer. If it’s more convenient, you can combine this and the
above part into one, e.g. if you wish to put everything in one Latex document, but this is
not required.
3. An R script file, i.e. with a .r extension containing your R code. This should be clearly
formatted, and include brief comments so that a reader can understand what it is doing.
The code should also be ready to run without any further modification by the user, and
should reproduce your results (approximately, for simulation-based results).
Please make sure that all required working, results, details of implementation and discussion are
contained in components 1 and 2 of the above list and not in the script file. The work will
be assessed based on the working, output and discussion in these components, and the script file
will only be used for verification of results. The exception is for the R code itself, whereby it is
sufficient to say “refer to script file” where a question asks you to write R code.
A complete submission consists of all the files in your final submission. The submission time of
the work will be based on the time at which the submission is complete, i.e. all files are uploaded.
Please carefully check after uploading your work that the files you upload are
the correct ones. Updates to any part of the submission after the deadline will be considered
a new submission and late penalties will be applicable.
Questions
1. Consider a random variable X with probability density function (pdf),
where 0 ≤ x ≤ 2π and r and R are parameters satisfying 0 < r < R.
(a) Describe the rejection algorithm using a uniform proposal distribution.
(b) Write an R function to implement your rejection method. The function should take
as inputs the parameters r and R, and also store the number of attempts needed to
successfully produce one accepted sample of X.
(c) For the case r = 0.9, R = 1, use your function to sample 10000 values of X. Plot a
histogram of your samples and overlay the true pdf, to check the sampler is working.
(d) How does the theoretical efficiency (in terms of expected number of attempts needed
to successfully produce one accepted sample of X) depend on the parameters r and
R?
(e) For various values of r and R, use your R function to illustrate empirically your answer
in (d).
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2. Data are available on the amount of a medicine in the blood (y) at n = 10 different times,
in hours, after administration (x):
x 1 2 3 4 5 6 7 8 9 10
y 1.3 1.9 2.4 2.5 2.4 2.5 2.6 2.7 2.7 2.7
Consider the nonlinear regression model
yi = b1 exp
{
1? exp
(
?xi
b2
)}
+ ?i, i = 1, . . . , n,
where b1 > 0 and b2 > 0 are parameters and the ?i ～ N(0, 0.01), i = 1, . . . , n, are
independent and identically distributed random errors. The times (x) were fixed and known
without error, so the data are viewed as a random sample of y for x fixed. Inference for the
unknown parameters b1 and b2 is required.
Here, we’ll perform a Bayesian analysis. To complete the model, we give b1 and b2
independent exponential prior distributions, each with parameter λ, where the pdf of the
exponential distribution is f(z) = λ exp(?λz). We’ll use λ = 0.01 for a diffuse prior on b1
and b2.
(a) Derive the posterior distribution π(b1, b2|x,y), up to proportionality in b1 and b2,
where x and y are the vectors of data.
(b) Give details of, and write R code to implement, the 2-d steepest ascent algorithm to
find the mode of the posterior distribution π(b1, b2|x,y). HINT: Work with the log
of the posterior!
NOTE: You are not expected to reparameterize to ensure b1 and b2 are positive —
you should start suitably close to the optimum and use a small enough step size so
that the algorithm converges to the optimum of interest.
2
(c) The marginal posterior distribution of b2 is
π(b2|x,y) =
∫ ∞
0
π(b1, b2|x,y)db1.
Give full details of Laplace’s method to compute π(b2|x,y) at a particular point b2.
(d) Write a function in R to compute π(b2|x,y) at a particular point b2 using Laplace’s
method derived in (c).
(e) Plot π(b2|x,y) using your function from (d).
(f) Write a function in R to perform the Golden-ratio method to find the mode of
p(b2|x,y), using your R function from part (d) as the function to optimize.
(g) Hence, find the mode of p(b2|y) to an accuracy of 1 decimal place.