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MATH5905 Statistical Inference
Assignment Two
Instructions: This assignment is to be completed collaboratively by a group of at most 3
students. The same mark will be awarded to each student within the group, unless I have good
reasons to believe that a group member did not contribute appropriately. This assignment
must be submitted no later than 11:59 pm on Sunday, 16 April 2023. The first page of the sub-
mitted PDF should be this page. Only one of the group members should submit the PDF file
on Moodle, with the names of the other students in the group clearly indicated in the document.
I/We declare that this assessment item is my/our own work, except where acknowledged, and
has not been submitted for academic credit elsewhere. I/We acknowledge that the assessor of
this item may, for the purpose of assessing this item reproduce this assessment item and provide
a copy to another member of the University; and/or communicate a copy of this assessment
item to a plagiarism checking service (which may then retain a copy of the assessment item on
its database for the purpose of future plagiarism checking). I/We certify that I/We have read
and understood the University Rules in respect of Student Academic Misconduct.
Name Student No. Signature Date
1
MATH5905 Term One 2023 Assignment Two Statistical Inference
Problem 1
Let X = (X1,X2, . . . ,Xn) be i.i.d. random variables, each with a density
f(x, θ) =
{
1
x
√
2π
e{−
1
2
[log(x)−θ]2}, x > 0
0 elsewhere
where θ ∈ R1 is a parameter. (This is called the log-normal density.)
a) Show that Yi = logXi is normally distributed and determine the mean and variance of
this normal distribution. Hence find E(log2Xi).
Note: Density transformation formula: For Y =W (X) :
fY (y) = fX(W
−1(y))|d(W
−1(y)
dy
| = fX(x)|dx
dy
|.
b) Find the Fisher information about θ in one observation and in the sample of n observa-
tions.
c) Find the Maximum Likelihood Estimator (MLE) of h(θ) = θ and show that it is unbiased
for h(θ). Is it also the UMVUE of θ? Justify your answer.
d) What is the MLE of h˜(θ) = 1/θ ? Determine the asymptotic distribution of the MLE of
h˜(θ) = 1/θ.
e) Prove that the family L(X, θ) has a monotone likelihood ratio in T =
∑n
i=1(logXi).
f) Argue that there is a uniformly most powerful (UMP) α−size test of the hypothesis
H0 : θ ≤ θ0 against H1 : θ > θ0 and exhibit its structure.
g) Using f) (or otherwise), find the threshold constant in the test and hence determine
completely the uniformly most powerful α− size test φ∗ of
H0 : θ ≤ θ0 versus H1 : θ > θ0.
Problem 2
Suppose that X is a random variable with density function
f(x, θ) =
1
β
e
− (x−θ)
β , θ < x <∞,
and zero else. Here β > 0 is a known constant and θ is an unknown location parameter
Let X = (X1, . . . , Xn) be a sample of n i.i.d. observations from this distribution.
i) Compute the distribution and density function for T = X(1).
ii) Find a statistic that has the MLR property.
iii) Justify the existence of a uniformly most powerful (UMP) α-size test of
H0 : θ ≥ θ0 versus H1 : θ < θ0.
2
MATH5905 Term One 2023 Assignment Two Statistical Inference
When β = 1, determine this test completely by calculating the threshold constant for
n = 5, θ0 = 2 and α = 0.05.
iv) Determine the power function of the UMP α test and sketch the graph of this function.
v) Suppose the following data was collected x = (1.1, 2, 1.3, 3.1, 1.65) and that β = 2. Test
the hypothesis that H0 : θ ≥ 1 versus θ < 1 with a significance level α = 0.05.
vi) Let Zn = n
(
X(1) − θ
)
. Show that the distribution of Zn does not depend on n and
recognize this distribution.
vii) Hence or otherwise justify that X(1) is a consistent estimator of θ.
Problem 3
Assume X1, X2, . . . , Xn are i.i.d. Bernoulli with parameter θ ∈ (0, 1), that is
Xi =
{
1 with probability θ
0 with probability (1− θ).
1. We want to test H0 : θ ≤ θ0 versus H1 : θ > θ0 at certain level α ∈ (0, 1). Justify the
existence claim of a uniformly most powerful (UMP) α test for this hypothesis testing
problem.
2. If n = 10 and θ0 = 0.27, show that the above UMP α = 0.05 size test randomly rejects
H0 with a probability of 0.28385 when
∑10
i=1Xi = 5.
Hint: You may use the R function dbinom to alleviate your calculations.
3. In 1000 tosses of a coin, 555 heads and 445 tails appear. How would you test the null
hypothesis of a fair coin against the alternative of a non-fair coin? Suggest a test that
has an asymptotic level α = 0.05 and defend your choice. Applying your test, answer
the question if it is reasonable to assume that the coin was fair.
Problem 4
Suppose X(1) < X(2) < X(3) < X(4) < X(5) are the order statistics based on a random sample
of size n = 5 from the standard exponential density f(x) = e−x, x > 0.
1. Find the numerical value of E(X(2)).
2. Find the density of the midrange M = 12(X(1)+X(5)). Your formula should only contain
a linear combination of exponential functions.
