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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