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


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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PART II (Third or Fourth Year)
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.
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]
please turn over
SECTION A continued
A5. An experiment consists of 6 units to which the following treatment combinations are applied.
(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]
please turn over
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
β 2
(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
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.
Using the identity link this data leads to ˆ
β= 1 and fitted values of 1.2,1.8,3.1,4.2.
Find the Fisher information numerically. [4]
please turn over
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
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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