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) = 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

1

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

η∈A+B. [2]

(d) Modify this matrix for the model in which the linear predictor η∈A+B+A.B. [2]

please turn over

2

SECTION B

B1. The exponential pdf with mean parameter µ > 0 is

f(y) = µ−1exp(yµ−1) 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

µ2µ∂µ

∂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 dµ

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

3

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

4

##### Document information

Uploaded by:
sawardekar_984

Views: 1156

Downloads :
0

University:
Amity University

Subject:
Mathematics

Upload date:
27/02/2013