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.

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Scan this pledge page and all the answer pages together and send to me at

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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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