ACST356 / ACST861 Mathematical Theory of Risk

This document discusses the mathematical theory of risk and explores the convergence of maximum values from different distributions as n goes to infinity. It covers the Fisher-Tippett-Gnedenko Theorem, the three classes of extreme value distributions, and the simulation of samples for fitting the GEV distribution.

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  

ACST356 / ACST861 Mathematical Theory of Risk 3
The Three Types Theorem (Fisher-Tippett, Gnedenko) asserts that if nondegenerate H
exists, it must be one of three types:
H(x) = exp(−e−x), all x (Gumbel)
(Fr´echet)
(Weibull)
In Fr´echet and Weibull, >α 0.
The three types may be combined into a single generalized extreme value (GEV)
distribution:
,
(y+ = max(y,0))
where μ is a location parameter, >ψ 0 is a scale parameter and ξ is a shape
parameter. ξ → 0 corresponds to the Gumbel distribution, >ξ 0 to the Fr´echet
distribution with α = 1/ξ, <ξ 0 to the Weibull distribution with α = −1/ξ. >ξ
0: “long-tailed” case, 1 − F(x) ∝ x−1/ξ,
ξ = 0: “exponential tail”
<ξ 0: “short-tailed” case, finite endpoint at μ − /ξ ψ
Question 2 (simulation)
Referring to Question 1 above, simulate 1,000 samples of M1,000 from the (a) Exponential(λ),
(b) Pareto(α,β) (Type I), and (c) Uniform(a,b) distribution. Choose your own parameter
values that are different to those in Question 1. Use maximum likelihood to fit the GEV
distribution to the simulated samples. Check the goodness-of-fit and also explain the results
under the context of extreme value theory.
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ACST356 / ACST861 Mathematical Theory of Risk 4
References
Castillo, E. (2012). Extreme value theory in engineering. Elsevier.
De Haan, L., & Ferreira, A. (2017). Extreme value theory: an introduction. Springer Science &
Business Media.
Galambos, J. (2014). Extreme value theory for applications. In Extreme value theory and
applications (pp. 1-14). Springer, Boston, MA.
Kotz, S., & Nadarajah, S. (2010). Extreme value distributions: theory and applications. World
Scientific.
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