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# statistic

Department of Mathematics and Statistics
4500-8500 – 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.
(iv) Pleadge: I pledge that I will complete this exam honouring the exam rules.
SIGNATURE:………………………………………. ……………………………………………….
Scan this pledge page and all the answer pages together and send to me at
or before 1.30 pm.
You must organize the pledge page and all answer pages as follows before scanning:
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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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