ACST885 Quantitative Methods

1
Assoc Prof Jackie Li 2019
ACST885 Quantitative Methods for Risk Analysis
2019 Assignment 2
Instructions
1. This is an individual assignment. Collaboration / plagiarism between students is strictly
prohibited.
2. You can type up your answers in Word or PDF, and / or write up and scan your answers.
The maximum allowable length is 10 pages. You should use R to perform simulations
and computations, and you must also submit the worked R files.
3. This assignment contains three questions. You must complete all the questions.
4. You must submit your work via Turnitin by Friday, 1st November 2019, 5pm. No
extensions will be granted (refer to the Unit Guide for details).
5. This assignment contributes 15% to the total assessment of this Unit. It will be assessed
on the basis of the accuracy of derivations and computations, reasonableness of analysis,
and quality of presentation.
Question 1 (extreme value theory)
Consider n random variables
X1 , X2 , … , Xn , which are independent and identically
distributed (iid) with cumulative distribution function (cdf) F x ( ) . Their minimum value is
expressed as U X X X n n = min( , , … , ) 1 2 . Under certain technical conditions, there exist a
sequence of positive numbers an and a sequence of numbers bn such that as n goes to infinity,
lim Pr( ) ( ) n n
n
n
U b
x H x
→ a

 =
,
in which H x ( ) is the cdf of the generalised extreme value (GEV) distribution for minima,
denoted as
GEV ( , , ) min    for -     and   0 . The limiting cdf H x ( ) must come
from one of the three classes: reversed Gumbel ( = 0), reversed Fréchet (  0), and Weibull
(  0), which are given below:
H x ( ) 1 exp( exp( )) x 


= – –
,  = 0, -    x ,
1
H x ( ) 1 exp( (1 ) )  x  



= – – – ,   0, x  

 +
,
2
Assoc Prof Jackie Li 2019
1
H x ( ) 1 exp( (1 ) )  x  



= – – – ,   0, x  

 + .
Deriving from first principles, find out to which class a minimum value from the (a)
Logistic(0,1), (b) Reversed Pareto(1,1), and (c) Exponential(1) distribution converges as n
goes to infinity. Identify clearly the values of  ,  , and  under each underlying distribution.
Note that the cdf of X ~ Logistic(μ,s) is F x ( ) (1 exp( )) = + – x-s -1 , the cdf of Y ~ Pareto(α,β) is
F y y ( ) 1 = –   – , and X ~ Reversed Pareto(α,β) comes from X = –Y.
Question 2 (simulation)
Referring to Question 1 above, simulate 10,000 samples of U10,000 from the (a) Logistic(μ,s),
(b) Reversed Pareto(α,β), and (c) Exponential(λ) distribution. Choose your own parameter
values that are different to those in Question 1. Use maximum likelihood to fit the GEV
distribution for minima to the simulated samples. Check the goodness-of-fit.
Question 3 (Bayesian theory)
Derive the posterior distribution for each of the three cases below under the Bayesian
framework. Show all your workings in detail.

(a) [beta]
[binomial]

( ) ( )
fprior p = ()+( ) p -1(1- p) -1 fprocess(x | p) =    nx    px(1- p)n-x
(b) ( )
fprior c-1 = ( )(c-1)- -1e- (c-1 )




 f process(x | c-1 )= c x -1 e-cx
[inverse gamma] [Weibull (with known  )]
(c) ( ) ( )
( )
 

  

=





1 2
2
2
prior
e
f ( )
( )
2
2
2
ln
2
process 2
1
| 

 




=
x
e
x
x
[gamma] [lognormal (with known  )]

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