ACST356 / ACST861 Mathematical Theory of Risk

ACST356 / ACST861 Mathematical Theory of Risk 1
ACST356 / ACST861 Mathematical Theory of Risk
2019 Assignment 2
By (Name of Student)
(Institutional Affiliation)
(Date of Submission)
1

ACST356 / ACST861 Mathematical Theory of Risk 2
Part A
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 maximum value is
expressed as Mn  max(X X1, 2, ... , Xn) . The Fisher-Tippett-Gnedenko Theorem states that
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( Mn bn  x)G x( ) ,
n an
in which G x( ) is the cdf of the generalised extreme value (GEV) distribution, denoted as
GEV( , , )  for    and 0. The limiting cdf G x( ) must come from one of the
three classes: Gumbel (0), Fréchet (0), and reversed Weibull (0), which are given
x  
G x( ) exp( (1 ) ),
0, x ,
x 
G x( )exp( (1  ) ), 0, x  . 
Deriving from first principles, find out to which class a maximum value from the (a)
Exponential(1), (b) Pareto(1,1) (Type I), and (c) Uniform(0,1) distribution converges as n
goes to infinity. Identify clearly the values of , , and  under each underlying distribution.
Suppose X1,X2,..., are independent random variables with the same probability
distribution, and let Mn = max(X1,...,Xn). Under certain circumstances, it can
be shown that there exist normalizing constants an > 0,bn such that
Pr .
2
below:
x
G x( )exp( exp ( )), 1
0,   x ,
   1
  
