LANCASTER UNIVERSITY
2007 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS
Math 352 Generalised Linear Models 90 minutes
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. Let Z∼Bino(m, µ), 0 < µ < 1, and fixed known integer m > 0. Define Y=Z/m so that
Y∼Binoprop(m, µ).
(a) State the values of Z(the support) which have non-zero probability. [2]
(b) Find the expectation and variance Y. [4]
(c) Write down the probability mass function of Y. [2]
(d) Explain why Zis not a GLM in the standard notation of GLMs but that Yis. [2]
A2. Define the logistic and logit functions, and show these are inverse. [8]
A3. Define the residual deviance of a GLM in terms of the log likelihood, expressed as a function
of the moment parameters, µi, and the observations, yi,i= 1,2,. . . , n. [4]
State what the deviance measures. [2]
A4. Show that the exponential family (EF) generated from the exponential pdf q(y) = 2exp(−2y),
y > 0, and Ymay be written as
f(y|θ) = 2 exp(−2y) exp{θy + log(1 −θ/2)}where θ < 2.[6]
Identify the mean of this pdf in terms of the canonical parameter θ. [4]
Find the maximum likelihood estimate of θbased on a single observation yfrom this pdf. [6]
please turn over
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