LANCASTER UNIVERSITY
2009 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATI C S & S TAT I S T I C S 2 Hours
Math 352: Generalized Linear Models
You should answer ALL Section A questions and ONE Section B question.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can
gain there is capped at 40.
SECTION A
A1. (a) Define an exponential family (EF) distribution of a random variable Ywith canonical
parameter θbased on a probability density or mass function q. You must clearly define
different quantities appearing in the definition. [7]
(b) Evaluate the log likelihood function, (θ), based on a single observation yfrom an EF
and find expressions for its first and second derivatives, θ(θ)andθθ (θ). [3]
(c) Use these results to find the mean and variance of Yfollowing the EF distribution in
terms of the derivatives of the cumulative generating function of q; you should state
carefully any general result about the score function that you use. [4]
(d) Show that the EF based on the probability mass function (pmf)
q(y)=1
2,y=0,1,
is given by
f(y|θ)= eθy
(1 + eθ),y=0,1.[3]
(e) Use (c) to find the relationship b etween the mean parameter and the canonical parameter
in this example. [1]
(f) Hence or otherwise derive the mean function and the variance function. [5]
please turn over
1
