# Values - Generalised Linear Models - Exam, Exams for Mathematics. Amity University

## Mathematics

Description: This is the Past Exam of Generalised Linear Models which includes Values, Fixed, Integer, Expectation, Variance, Probability Mass Function, Standard Notation, Logit Functions, Logistic etc. Key important points are: Values, Fixed, Integer, Expectation, Variance, Probability Mass Function, Standard Notation, Logit Functions, Logistic, Residual Deviance
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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 ZBino(m, µ), 0 < µ < 1, and ﬁxed known integer m > 0. Deﬁne Y=Z/m so that
YBinoprop(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) = 2 exp(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]
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SECTION A continued
A5. An experiment consists of 6 units to which the following treatment combinations are applied.
1A2B2
2A1B2
3A1B2
4A2B1
5A3B1
6A3B1
(a) Write out the matrix that corresponds to this design in terms of the indicator vectors
for these factor levels. [4]
(b) Deﬁne the factor Ain terms of these indicator vectors. [2]
(c) Write out the design matrix (the Xmatrix) for the model in which the linear predictor
ηA+B. [2]
(d) Modify this matrix for the model in which the linear predictor ηA+B+A.B. [2]
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SECTION B
B1. The exponential pdf with mean parameter µ > 0 is
f(y) = µ1exp(yµ1) for y > 0.
A one dimensional covariate xis associated with the observation ythrough the unspeciﬁed
link function gwhere g(µ) = η, the linear predictor, and η=βx. Consider estimating the
regression parameter βfrom a single observation.
(a) Write down the log-likelihood function for βand, by using the chain rule, ﬁnd the score
function and the observed information for β. [8]
(b) Show that the Fisher information for βis
1
µ2µµ
β 2
.[6]
(c) Three possible candidates for the link function are the identity, log and reciprocal links
given by
reciprocal: 1
µ=βx; log: log(µ) = βx; identity: µ=βx.
Explain how these three links lead to diﬀerent interpretations of the parameter βby
computing
dx . [6]
(d) Comment on the relative merits of these three links. [6]
(e) Suppose n= 4 independent observations are made on this GLM at the xpoints 1,1,2,2
resulting in yvalues of 1,2,3,4 respectively.
β= 1 and ﬁtted values of 1.2,1.8,3.1,4.2.
Find the Fisher information numerically. [4]
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SECTION B continued
B2. There are three notations for GLMs: generic, index, vector notation.
(a) Explain these diﬀerent notations when describing the response variable, [6]
(b) The diagram of a GLM is
m
m
m m
m
m
J
J
J
Add labels to the diagram to represent the generic concepts that go to deﬁne a GLM. [4]
(c) Give brief deﬁnitions of these generic concepts. [8]
(d) Add directions to the edges of this diagram to help explain the interrelationships be-
tween the generic concepts of a GLM, and label the edges with GLM functions where
appropriate. [4]
(e) Describe these interrelationships. [8]
end of exam
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