UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH3821 Statistical Modelling and Computing
Midsession Test, T2 2021
Time allowed 60 Minutes
Total number of questions 2
Where the question is a multiple choice, place your selection (a, b,…) in the answer box provided. You
can choose more than one.
Where th question is not a multiple choice, provide the answer in the form specified in the answer box.
When you finish, you may download the notebook and submit as .ipynb file, or a pdf file.
I have provided space at the end of the test for you to provide your answers in a concise format.
Student Name
Student ID
Question 1 [Total 9 marks, 3 parts]
Let be a random variable with values in (or or ), and suppose its probability distribution
depends on a single parameter . Then it belongs to the exponential family if it admits the form
for some (measurable) functions and .
Another way of writing this is
where and .
Identify if the following probability density functions belong to the exponential family. If yes, is the
distribution of canonical and what is the natural parameter of this distribution.
a). [3 marks] For the Pareto distribution . Which of the following statement(s) are true.
In [ ]:
In [ ]:
In [ ]:
In [ ]:
Y R N
0
Z f(y; θ)
θ
f(y; θ) = s(y)t(θ)e
a(y)b(θ)
a, b, s, t
f(y; θ) = exp[a(y)b(θ) + c(θ) + d(y)]
s(y) = e
d(y)
t(θ) = e
c(θ)
Y
f(y; θ) = θy
θ 1
(a) Does not belong to the exponential family of distributions;
(b) Belongs to the exponential family of distributions and is of cannonical form with natural parameter
;
(c) Belongs to the exponential family of distributions but is not of cannonical form;
(d) Belongs to the exponential family of distributions and is of cannonical form with natural parameter
.
Answer
b). [3 marks] For the Exponential distribution . Which of the following statement(s) are
true.
(a) Does not belong to the exponential family of distributions;
(b) Belongs to the exponential family of distributions and is of cannonical form with natural parameter
;
(c) Belongs to the exponential family of distributions but is not of cannonical form;
(d) Belongs to the exponential family of distributions and is of cannonical form with natural parameter ;
Answer
c). [3 marks] For the Negative binomial distribution ( is known) . Which of
the following statement(s) are true.
(a) Does not belong to the exponential family of distributions;
(b) Belongs to the exponential family of distributions and is of cannonical form with natural parameter
;
(c) Belongs to the exponential family of distributions but is not of cannonical form;
(d) Belongs to the exponential family of distributions and is of cannonical form with natural parameter
.
Y
Y
(θ+ 1)
Y
Y
2(θ+ 1)
In [ ]:
In [ ]:
In [ ]:
f(y; θ) = θe
yθ
Y
Y
θ
Y
Y θ
In [ ]:
In [ ]:
In [ ]:
r f(y; θ) = ( )θ
r
(1 θ)
y
y+r 1
r 1
log(1 θ)
1 θ
Answer
Question 2 [Total 14 marks, 7 parts]
The data below are times to death, , in weeks from diagnosis and (initial white blood cell count), ,
for seventeen patients suffering from leukemia.
a). [1 mark] From an appropriate visualisation of the data, do the data show any trend
(a) no trend in the data;
(b) when increases also increases approximately linearly;
(c) when increases decreases approximately exponentially.
Answer
b). [2 marks] A possible specification for is
which will ensure that is non negative for all values of the parameters and all values of . Which link
function is appropriate in this case
(a) ;
(b) ;
(c) identity function.
In [ ]:
In [ ]:
In [ ]:
In [ ]:
y
i
log
10
x
i
In [1]: y=c(65,156,100,134,16,108,121,4,39,143,56,26,22,1,1,5,65)
x=c(3.36,2.88,3.63,3.41,3.78,4.02,4.00,4.23,3.73,3.85,3.97,4.51,4.54,5.00,5.00,4.72 ,5.00)
x y
x y
In [ ]:
In [ ]:
In [ ]:
E(Y )
E(Y
i
) = exp(β
0
+ β
1
x
i
)
E(Y ) x
exp
log
Answer
c). [2 marks] The exponential distribution is often used to describe survival times. The probability
distribution is
You would like to fit a model with the equation for given by
and the exponential distribution using glm() in R. What is the family object set to in order to fit such model
(a) family=Gamma(link=”log”);
(b) family=Gamma;
(c) family=gaussian(link=”exp”);
(d) family=Gamma(link=”exp”);
(e) family=binomial(link=”log”);
(f) family=gaussian(link=”log”).
As a hint see the following extract from the object family documentation in R
family(object, …)
binomial(link = “logit”)
gaussian(link = “identity”)
Gamma(link = “inverse”)
inverse.gaussian(link = “1/mu^2”)
poisson(link = “log”)
quasi(link = “identity”, variance = “constant”)
quasibinomial(link = “logit”)
quasipoisson(link = “log”)
Answer
In [ ]:
In [ ]:
In [ ]:
f(y; θ) = θ exp( yθ).
E(Y
i
)
E(Y
i
) = exp(β
0
+ β
1
x
i
)
In [ ]:
In [ ]:
In [ ]:
d). [2 marks] You have now fitted the model with the equation for given by
and the exponential distribution using glm() in R and obtained the following result
Error in parse(text = x, srcfile = src): :2:10: unexpected symbol
1:
2: Deviance Residuals
^
Traceback:
What are the parameter estimates and are they significant assuming that
(a) and and they are both significant;
(b) and and they are both significant;
(c) and and is not significant;
(d) and and they are both significant;
(e) and and is not significant.
Answer
e). [2 marks] What is the 95% confidence interval for the parameter in the model
