Department of Mathematics and Statistics4500-8500 – Generalized Linear Models
Final Exam
December 5, 2022
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Final Exam Rules:
(i) Exam is closed notes, closed book, just like regular in-person exams.
(ii) There are 5 questions. Students in 4500 answer questions 1, 2, 3 & 5.Students in 8500 answer all questions. Normal statistical tables have already been
sent.
question 1: /25
question 2: /20
question 3: /20
question 4: /10
question 5: /15
………………………………………………………..
Answer each question in separate pages.
(iii) Start writing the exam at 10 am. At 1.00 pm or before stop writing.
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Ans to Question 1
and so
1
Ans to Question 5
2
1. (25 marks) Consider a random sample (T1 /n1 , T2 /n2 , . . . , Tk /nk ), where
Ti ∼ binomial(ni , pi ). Express the distribution of Y = T /n in the form
of an exponential family of distributions. Find the canonical parameter
and hence find the mean and variance
of y. Use canonical link and the
P
systematic component as η =
βj Xj and show that the score ∂l/∂βj
(j = 1, 2, . . . , p) is
k
X
uj =
ni (yi − pi )xji
i=1
Show that solving uj = 0 for j = 1, 2, . . . , p the Newton Rhaphson method
and Fisher’s scoring method are equivalent and hence show that estimate
of β by either method is equivalent to the Weighid least squares estimate
of β by regressing
(y − p̂)
onX
ẑ = X B̂ +
p̂(1 − p̂)
with weight Ŵ = np̂(1 − p̂), whereˆdenotes estimates at the k th iteration.
For the above model obtain an expression for deviance and generalized
Pearson statistic and hence suggest two estimates for possible over dispersion in the data.
2. (20 marks)
a)Let Y have a probability mass function
y+r−1 r
y
r−1 θ (1 − θ) ,
f (y; θ) =
where r is known. Show that the distribution belongs to the exponential
family and hence find the mean and variance of y.
b) Consider a 2 × 2 contingency table with one margin fixed at n1 and n2
y1
y2
n 1 − y1
n 2 − y2
n1
n2
y1 ∼ binomial(n1 , p1 )
y2 ∼ binomial(n2 , p2 )
Ψ=
p1 (1 − p2 )
p2 (1 − p1 )
t
The conditional distribution of y1 given T = t, n1 and n2 (note T = y1 +y2 )
is
n2 y λ
n1
1
y1 t−y1 e
P (Y1 = y1 | n1 , n2 , t) = P n1 n2 sλ
s s
t−s e
Show that the conditional distribution of y1 belongs to the exponential
family of distributions. Find the canonical parameter. Hence find the
mean and the variance of y1 .
3. (20 marks)
Let Y1 , Y2 , . . . , Yn be independent poisson random variables with means
µ1 , µ2 , . . . , µn . Associated with each Yi is a covariate vector, Xi , of length
p. Show that νi = log µi is the natural parameter of the poisson distribution.
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(a) Show that T = X ′ Y is a vector of sufficient statistic for β.
(b) Find the score vector for β.
(c) Find the observed and expected information matrix for β and hence
show how to obtain the MLE for β.
4. (10 marks)
The probability function of discrete random variable y is
Γ(y + c−1 )
P (Y = y) =
y! Γ(c−1 )
cm
1 + cm
y
1
1 + cm
c−1
.
This distribution, denoted by NB(m, c), is called a negative binomial distribution with mean m and dispersion parameter c. Suppose y1 , y2 , . . . , yn ∼
NB(m, c). Show how you would find maximum likelihood estimates of m
and c. Do you get explicit solution for any of m or c?. If explicit estimate
for any parameter does not exit, show how you would find it iteratively.
5. (15 marks)
Consider the data given below. The mle of π and θ are π̂ = 0.0776 and
θ̂ = 0.0252
x/n : 0/5, 2/6, 0/7, 0/7, 0/8, 0/8, 0/8, 1/9, 2/9, 1/10
(i) By using the score test is there any evidence of over dispersion in the
data?
(ii) using θ̂ find an estimate of σ 2 , where σ 2 is the parameter used in
GLM to represent over-dispersion in binomial data.
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