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日期：2023-04-13 10:13

MATH5905 Term One 2023 Assignment Two Statistical Inference

University of New South Wales

School of Mathematics and Statistics

MATH5905 Statistical Inference

Term One 2023

Assignment Two

Given: Monday 3 April 2023 Due date: Sunday, 16 April 2023

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

[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) :

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.