The University of New South Wales
Department of Statistics
MATH5855 - Multivariate Analysis I
Assignment 2
Due Tuesday, 25th September 2018, 5pm
1. i) You are asked to write a subroutine (module) within SAS/IML with an input:
an arbitrary data matrix with n datapoints, each containing p dimensions (p 0 is an unknown
scalar and D is a known symmetric positive de nite matrix. Show that the Maximum
Likelihood Estimator of is ^ = 1nptr(D 1B) where B = Pni=1xix0i. Show also that
np^
2
np: Hence suggest a two-sided con dence interval for at level (1 ):
(Hint: You may nd it useful to consider vectors Yi = D 1=2Xi)
2
5. For a random vector (X;Y)0 of continuous random variables with marginal distri-
butions F and G; the coe cient of upper dependence is de ned as
upper = limu!1P(Y >G 1(u)jX >F 1(u))
provided that the limit exists. In the context of copulae, this results in the investigation
of
upper = limu!1(1 2u+C(u;u))=(1 u):
When upper2(0;1] we say that there exists an asymptotic dependence in the upper tail;
when upper = 0 the random variables are said to be asymptotically independent in the
upper tail.
Show that the Gumbell-Hougaard copula
C (u;v) = exp( [( logu) + ( logv) ]1= );u2[0;1];v2[0;1]
with a parameter 2[1;1) exhibits upper tail dependence when > 1):
(Reminder: as we know when = 1 the above copula coincides with the independence
copula).