MA318-6-AU

UNIVERSITY OF ESSEX

Final Year Examinations 2020


STATISTICAL METHODS


Time allowed: 24 hours

You are not expected to spend 24 hours on this paper, this time is to take into account personal
circumstances. You do not need to complete this paper in a single sitting. You may complete the paper at
the time(s) of day that best suit you, provided you submit by the deadline.

We expect students to be able to complete this paper within 3 hours.

IMPORTANT
If you intend to apply for IFoA exemptions you MUST submit your answers within 4 hours of the start
of the exam. FASER time stamps your submission and we will verify to the IFoA your time of
submission.

Please see your exam timetable or check on FASER for the deadline by which to upload your answers.

If you are normally allowed either additional writing time or a rest break this will not apply in this case as the
24 hours given to all students is considered sufficient. If you have any questions relating to this please
contact the Student Wellbeing and Inclusivity Team. The exception to this is if you are applying for IFoA
exemptions and you would then have your additional time on top of the standard exam length of 3 hours,
plus 1 hour.

The times shown on your timetable are in British Summer Time (BST). Please check online for a
conversion to your local time if you will be undertaking your assessment outside the United
Kingdom

Number of questions: 9

Equipment needed: Useful distribution densities at the end of this question paper

Questions are NOT of equal weight

Candidates must attempt ALL questions.

Contacts
If you believe there is an error in this paper please email maths@essex.ac.uk

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Academic Offences
Please do not communicate with any other candidate in any way during this assessment. Your response
must be your own work. Procedures are in place to detect plagiarism and collusion.

Submitting your answers
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Any late submissions will receive a mark of zero.
Additional instructions on completing and submitting your exam

Format of your answers
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these will be marked anonymously.
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uploading your answers, you can email them to maths@essex.ac.uk with [MA318 exam] in the
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FASER.

 Only your latest submission (within the deadline) will be marked.

Extenuating circumstances
If you are unable to sit this exam, you must complete a Notification of Absence form and clearly
state the reasons why you are unable to submit the work.

We know that students are working under different conditions at the moment. If you encounter any
difficulties with your exam that you believe has affected your performance you can inform the
Exam Board of this by submitting an Extenuating Circumstances form
https://www1.essex.ac.uk/students/exams-and-coursework/ext-circ.aspx

Academic Offences
You are reminded that copying from textbooks, articles or any online or offline sources is
plagiarism, which is an academic offence. Collaborating with other students is collusion, which is
also an academic offence. Any student suspected of committing an academic offence will be
interviewed in person to determine whether the work they have submitted is entirely their own.
More information about academic offences can be found at
https://www.essex.ac.uk/student/exams-and-coursework/about-academic-offences
2 MA318-6-AU
1. The table below shows the increment claims paid on a portfolio of insurance policies. Claims are
fully paid by the end of development year 3.
Policy year / Development year 0 1 2 3
2008 (year 1) 100 35 30 10
2009 (year 2) 89 20 16
2010 (year 3) 112 33
2011 (year 4) 132
We assume that the above random values are independent and from Poisson(αiβj), with αi, βj > 0,
where i is the policy year index and j is the development year index.
Use the chain ladder algorithm to estimate the values αi and βj .
[10 marks]
2. Claim amounts on a certain type of insurance policy follow an exponential distribution with
mean=5, i.e. it has probability density
f(x) = 0.2e−0.2x, x > 0.
The insurance company purchases a special type of reinsurance policy so that for a given claim X
incurred by insurer, the reinsurance company pay
0 if 0 < X < 3
0.5X − 1.5 if 3 ≤ X < 10
X − 6.5 if X ≥ 10
to the insurance company.
Calculate the expected amount paid by the reinsurance company on a random chosen claim incurred
by insurer.
[12 marks]
3. A car insurance company has a particular policy. Denote N as the number of claims per annum and
we assume that N follows a Poisson distribution with parameter µ. Claim amounts (X1, · · · , XN )
are assumed to be independent from each other, with exponential distribution λe−λx. Let
S =
∑N
i=1Xi denote the aggregate annual claims from this policy. A check is made for ruin only at
the end of the year.
Suppose that the premium income per year is c.
(a) Derive the moment generating function of N , Xi and S. [9 marks]
(b) Suppose that we can obtain the estimates for the mean and variance of S and denote them as
m1 and m2, respectively. If we use a normal approximation to the distribution of S, briefly
explain how to calculate the initial capital, U , required in order that the probability of ruin at
the end of the first year is 0.01. [3 marks]
3 MA318-6-AU
4. (a) Determine the Jeffreys non-informative prior for the parameter β of a distribution function with
density
f(x|α, β) ∝ βαxα−1e−βx
where α is supposed to be given.
[8 marks]
(b) Explain in what sense Jeffreys prior is non-informative.
[2 marks]
5. Assume that we have an independent sample X1, X2, · · · , Xn from a geometric distribution with
parameter p, where the mass function is:
f(x|p) = p(1− p)x, 0 < p ≤ 1, x = 0, 1, 2, ...
The prior density function for p is given by
f(p) =
Γ(8)
Γ(5)Γ(3)
p4(1− p)2, 0 < p < 1
a. Find the posterior distribution for p. [4 marks]
b. Show that, under the prior distribution, E
(
1−p
p
)
= 3
4
. [8 marks]
6. Suppose the weather of a Sunday can be one of three possibilities, θ1 (rainy), θ2 (windy) or θ3
(sunny) and a person can choose one of the three actions to do on that day a1 (picnic), a2 (shopping)
or a3 (waching football match). Through consideration, he arrives at the following loss matrix
a1 a2 a3
θ1 4 1 2
θ2 0 1 5
θ3 1 2 0
Suppose an action δ of the person must have the form δ = p1 < a1 > +p2 < a2 > +p3 < a3 >,
which means with probability pk the person takes action ak, k = 1, 2, 3.
Find the best δ based on the minimax rule.
[15 marks]
7. Suppose that we observe one data point x ∼ N(θ, σ2) where σ2 is known. The prior distribution is
θ ∼ N(0, 1). Find the (prior) predictive distribution.
[12 marks]
4 MA318-6-AU
8. The waiting time for a bus at a given station at a certain time of a day is known to have a uniform,
U(0, θ), distribution. It is desired to test H0 : 0 < θ < 12 versus H1 : θ > 12. From other similar
routes, it is known that θ has a Pareto distribution P(5, 3). If the waiting times 10, 3, 2, 5, 8 are
observed, calculate the posterior probability of each hypothesis and the Bayes factor.
(You are given the following information:
The Pareto P(x0, α) distribution has a pdf,
f(θ|x0, α) = α
x0
(x0
θ
)α+1
I(x0,∞)(θ)
with θ > x0, 0 < x0 <∞ and α > 0. It has mean αx0/(α− 1) if α > 1.)
[12 marks]
9. Suppose we obtain a posterior which is a Beta(5,8) distribution. The following R code estimates the
mean of this posterior distribution. Explain each step of the following R code and explain why it
provides a consistent estimate for the mean of this posterior.
n = 1000
alpha = 5
beta =8
x = rbeta(n, alpha, beta)
mean(x)
[5 marks]
END OF PAPER
MA318 – useful distribution densities
1 p follows a Beta-distribution, Beta(; ), then it has probability density function (pdf)
f(pj; ) = 1
B(; )
p1(1 p)1; p 2 [0; 1]; ; > 0
with mean =( + ). Here
B(; ) =
()()
(+ )
and () has the following property
( + 1) = ():
2 p = (p1; ; pk) follows a Dirichlet-distribution, Dirichlet(1; ; k), then it has pdf
f(p1; ; pkj1; ; k) = (
P
i i)
(1) (k)p
11
1 pk1k ;
X
pi = 1; pi 2 [0; 1]; i > 0
with mean k= (
P
i i) for pk.
3 X follows a Binomial-distribution, then it has pdf
f(xjn; p) =

n
x

px(1 p)nx; p 2 [0; 1]
with mean np.
4 follows a Normal distribution of mean and variance 2, then it has pdf
f(j; 2) = 1p
22
e
()2
22
2 MA318 – useful distribution densities
5 X follows a Negative binomial (NB) distribution, NB(r; p), then it has pdf
f(xjr; p) =

x+ r 1
x

(1 p)rpx
Note that the NB random variable X means, in a sequence of i.i.d. Bernoulli trials, the number of
successes before r failures occur.
6 X = (X1; ; Xk) follows a multinomial distribution,M(n;p), then it has pdf
f(xjn;p) = n!
x1! xk!p
x1
1 pxkk
where
P
i xi = n, pi 2 [0; 1] and
P
i pi = 1.
7 follows a Gamma distribution, Gamma(; ), then it has pdf
f(j; ) =

()
1e; ; > 0
Note that if = 1, the gamma distribution becomes an exponential distribution.
8 X follows a uniform distribution, Uniform(a; b) (or using notation U [a; b], uniform[a; b]), then it has
pdf
f(xja; b) = 1
b a; for all x 2 [a; b];
otherwise f(xja; b) = 0.
9 X follows a Poisson(), then it has probability mass function
p(xj) =
x
x!
e;