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MTH731U /MTHM731 /MTH731P: Computational Statistics

创建日期：2024-05-06 18:16:39

所属分类：数学mathematics

标签： MTH731U

Computational Statistics

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MTH731U /MTHM731 /MTH731P: Computational
Statistics
Duration: 3 hours
Apart from this page, you are not permitted to read the contents of this question paper
until instructed to do so by an invigilator.
You should attempt ALL questions. Marks available are shown next to
the questions.
Only non-programmable calculators that have been approved from the college list of
non-programmable calculators are permitted in this examination. Please state on your
answer book the name and type of machine used.
Complete all rough work in the answer book and cross through any work that is not to be
assessed.
Possession of unauthorised material at any time when under examination conditions is an
assessment offence and can lead to expulsion from QMUL. Check now to ensure you do not
have any notes, mobile phones, smartwatches or unauthorised electronic devices on your
person. If you do, raise your hand and give them to an invigilator immediately.
It is also an offence to have any writing of any kind on your person, including on your body. If
you are found to have hidden unauthorised material elsewhere, including toilets and
cloakrooms, it shall be treated as being found in your possession. Unauthorised material
found on your mobile phone or other electronic device will be considered the same as being in
possession of paper notes. A mobile phone that causes a disruption in the exam is also an
assessment offence.
Exam papers must not be removed from the examination room.
Examiners: J. Griffin, H. Maruri-Aguilar
© Queen Mary University of London (2018) Turn Over
Page 2 MTH731U /MTHM731 /MTH731P (2018)
Question 1. [9 marks]
Consider the following lines of R code:
v = c(1.4,0.7,0.2,1.5,-1.9,2.2,-0.8,1.3,1.1,0.6)
n = length(v)
p = ((1:n)-0.5)/n
q = qnorm(p)
plot(q, sort(v))
(a) What type of outcome does this code produce? [3]
(b) What values will p contain after the third line has been executed? [3]
(c) Explain the meaning of the command q = qnorm(p). [3]
Question 2. [21 marks]
(a) Define the Wilcoxon signed-rank statistic W+ for a sample z1, . . . ,zn. [3]
(b) What would be the null hypothesis for the Wilcoxon signed-rank test? Show that under
this null hypothesis for a sample of size n, W+ has a mean
E(W+) =
n(n+ 1)
4
. [8]
(c) Blood cholesterol measurements were taken for eight patients before and after a course
of medication.
Patient 1 2 3 4 5 6 7 8
Before 5.7 6.2 7.4 6.0 5.1 7.3 6.9 6.4
After 6.5 4.1 5.0 6.3 4.2 3.3 5.2 3.8
We want to find out if the medication has led to a decrease in the cholesterol level. Use
an appropriate rank test to test this hypothesis at the 5% level of significance. [10]
Question 3. [15 marks]
(a) Consider the samples 6.8,5.3,7.1 and 4.2,5.9 from two populations. We want to know if
the mean in the population associated with the first sample is different from the
population mean for the second sample. We are not prepared, however, to assume that
the data are normally distributed.
Suppose we want to perform a permutation test. State an appropriate null hypothesis
and a test statistic. Perform a permutation test at the 10% significance level to test the
hypothesis. [12]
(b) For sample sizes which are too large to calculate the exact null distribution, even by
computer, explain how we might approximate the null distribution for a permutation
test. [3]
© Queen Mary University of London (2018)
MTH731U /MTHM731 /MTH731P (2018) Page 3
Question 4. [20 marks]
(a) Consider the following data:
1.7, 5.9, 7.2, 2.8, 5.1, 5.6, 6.4, 4.4, 2.5, 4.6
Find the histogram estimator of the probability density function fˆH(y) for all y ∈,
taking the interval end points to be 0,2,4,6,8. [8]
(b) For a given sample size, how do the bias and variance of the histogram estimator fˆH(y)
at a single point y change as the interval width decreases? [4]
(c) State the general formula for a kernel density estimator (KDE) of a probability density
function, explaining all terms. Which component of a KDE has the strongest influence
on the appearance of the estimated density? [5]
(d) State one advantage of using a KDE rather than a histogram for estimating a probability
density. [3]
Question 5. [16 marks]
(a) Suppose that we have two random samples x = x1, . . . , xm and y = y1, . . . ,yn, which are
assumed to be independent of each other. Let θ be the quantity we are interested in, the
ratio of the standard deviation of the first population and the the standard deviation of
the second population, which we estimate using
θˆ =
√
Var(x)
Var(y)
Describe how we would generate a non-parametric bootstrap sample for θˆ, and how we
would use this sample to estimate the standard error of θˆ. [9]
(b) Explain how we would calculate a 95% percentile confidence interval for θˆ using the
bootstrap sample in (a). [4]
(c) In general, if a confidence interval (θL, θU) is calculated for a population parameter θ,
define what is meant by the coverage of the confidence interval. [3]
© Queen Mary University of London (2018) Turn Over
Page 4 MTH731U /MTHM731 /MTH731P (2018)
Question 6. [19 marks]
Suppose that our data is of the form (y1, x1), . . . , (yn, xn). We wish to fit models of the form
E(Y) = g(x;β), where β is a vector of parameters to be estimated.
(a) Describe the general procedure for using leave-one-out cross-validation to obtain a set
of predictions yˆ[1], . . . , yˆ[n]. [5]
(b) Define the predicted residuals that result from the leave-one-out cross-validation
procedure, and define the PRESS statistic. [4]
(c) Suppose that g depends on a set of spline functions and we estimate β by minimizing
the penalized sum of squares
PSS =
n∑
i=1
(yi−g(xi;β))2 +λ
∞∫
−∞
g′′(x;β)2 dx
where λ > 0 is a smoothing parameter.
The answers do not need any details about spline functions.
(i) Explain why for sufficiently large values of λ, the fitted model approaches a linear
function. [3]
(ii) Suppose that we have fitted this model for a range of values of λ and calculated the
PRESS statistic each time, with the results as plotted below.
15
20
25
30
35
PR
ES
S
0.0 0.5 1.0
l
Explain how this graph would be used to select a value of λ. By doing this, what
feature of the fitted model are we selecting for? Why would PRESS initially
decrease as λ increases for small values of λ? [7]
End of Paper – An appendix of 2 pages follows.
© Queen Mary University of London (2018)
MTH731U /MTHM731 /MTH731P (2018) Page 5
Appendix: statistical tables
Normal distribution function
Table 1: The standard normal cumulative distribution function Φ(x) for the given values of x.
The cdf for x< 0 can be found using the fact that Φ(x) = 1−Φ(−x). For x≥ 3.8, 1−Φ(x)< 10−4.
x Φ(x) x Φ(x) x Φ(x) x Φ(x) x Φ(x)
0.0 0.500 0.8 0.788 1.6 0.945 2.4 0.992 3.2 0.9993
0.1 0.540 0.9 0.816 1.7 0.955 2.5 0.994 3.3 0.9995
0.2 0.579 1.0 0.841 1.8 0.964 2.6 0.995 3.4 0.9997
0.3 0.618 1.1 0.864 1.9 0.971 2.7 0.997 3.5 0.9998
0.4 0.655 1.2 0.885 2.0 0.977 2.8 0.997 3.6 0.9998
0.5 0.691 1.3 0.903 2.1 0.982 2.9 0.998 3.7 0.9999
0.6 0.726 1.4 0.919 2.2 0.986 3.0 0.999 3.8 0.9999
0.7 0.758 1.5 0.933 2.3 0.989 3.1 0.999
Table 2: Selected upper quantiles of the standard normal distribution. For each p in the first
row, the second row contains the value of x such that Φ(x) = 1− p.
p 0.1 0.05 0.025 0.01 5×10−3 10−3 5×10−4 10−4
Φ−1(1− p) 1.28 1.64 1.96 2.33 2.58 3.09 3.29 3.72
Wilcoxon signed-rank critical values
Table 3: Lower critical values of the one-sample Wilcoxon signed-rank statistic W+ for samples
of size n. For each n and P, the entry in the table is the largest value x such that P(W+ ≤ x) ≤ P.
If there is no such x, then the entry is blank (-).

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