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ACST8040 Quantitative Research Methods

Assignment 2
(Due 6pm on Friday, 6 May 2022)

Instructions:
• This test consists of 5 problem-solving questions requiring detailed solutions.
• It is to be completed independently by each student.
• It will count for 40% of assessment.
• Work out the details and show the steps to solve each problem, including the right theory
and methods used, appropriate formulae to calculate the answers, as well as the steps of
calculations.
• Each question carries 8 marks.
• The full of the assignment is 40.
• Submit your answers in PDF file via Turnitin on iLearn by Friday 6pm, 6 May 2022.
• The submitted answers must be typed (not handwritten).
• Note that Turnitin requires at least 20 typed words to submit a file.

Rules for use of R programme:
• If a question indicates to use R, present relevant R-codes and outputs in your answers.
• It suffices to copy-paste R-codes and outputs in your answer sheets, and not necessary
to provide R-scripts.
• Use your own words to write answers obtained from R (not just provide R-scripts).
• For any question (or part of a question) with no mention of using R, your submitted
answers should not rely on R.

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Question 1 [8 marks]
Let X be a random variable with E[ ]X < ∞ .
(a) Derive the following formula of E[ ]X for continuous X :
[ ]E[ ] ( ) 1 ( )
r
r
X r F x dx F x dx
∞
−∞
= − + −∫ ∫ for any real number r∈ ,
where ( )F x is the cumulative distribution function (cdf) of X .
(b) Prove that if X is symmetric about a∈ , then a is a median of X and E[ ]X a= .
Note: The formula of E[ ]X in part (a) is valid for non-continuous X as well.
(c) Determine if the converse of part (b) is also true (if a is a median of X and E[ ]X a= ,
then X is symmetric about a ). If so, prove it; if not, present a counterexample.


Question 2 [8 marks]
An investor recorded the investment returns (profits in $000) from 14 assets as follows:
(240,408,217, 40,113, 60,312, 15,40,60,375,40, 124,180)− − − −
Consider the profits as the observed values of independent continuous random variables
1 14, ,X X with a common median θ .
Carry out the following statistical analyses:
(a) Find the exact p-value of testing the null hypothesis 0 : 0H θ = against the alternative
1 : 0H θ > (positive profits) by the sign test.
(b) Estimate the median θ and obtain its exact confidence interval at the level nearest 95%
based on the sign statistic.
(c) Calculate the p-value of testing 0 : 0H θ = against 1 : 0H θ > by the Wilcoxon signed
rank test and normal approximation.
(d) Estimate the median θ and obtain its approximate 95% confidence interval based on the
Wilcoxon signed ranks.
(e) Compare and comment on the p-values of the tests and the confidence intervals of θ
based on the sign statistic and the Wilcoxon signed ranks in parts (a)-(d).

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Question 3 [8 marks]
Observations of two random samples are given below:
1 8( , , ) (5,15,8,10, 7,7,12,5)X X = − , 1 4( , , ) ( 2,5, 2,1)Y Y = − −
(a) Under the location-shift model for the two samples with location shift ∆ , calculate the
exact p-value of the Wilcoxon rank sum test for testing 0 : 0H ∆ = against 1 : 0H ∆ <
conditional on any observed ties by enumeration.
(b) Let C denote the Ansari-Bradley test statistic for the two-sample dispersion problem.
Find the observed value c of C based on the above two samples.
(c) Assume the location-scale model for the two samples. Calculate the exact probability
Pr( )C c= conditional on observed ties under 0 : Var( ) Var( )H X Y= by enumeration.


Question 4 [8 marks]
The following two samples of data are extracted from a study:
X = (9.0, 6.6, 7.4, 9.0, 6.0, 7.2, 7.0, 7.7, 7.3, 8.3, 9.5, 7.7, 8.6, 8.4, 10.8, 8.8, 8.6, 6.8, 8.2, 7.5)
Y = (9.3, 14.0, 9.8, 9.6, 9.3, 13.1, 13.7, 12.3, 8.2, 8.9, 5.0, 11.7, 8.6, 7.7, 10.1, 11.3, 9.9, 12.8,
12.8, 11.1, 11.8, 8.9, 11.5, 10.6, 4.0)
(a) Use R to carry out the following five tests (ignore warning messages for ties):
• Wilcoxon rank sum test of equal location between X and Y
• Ansari-Bradley test of equal dispersion between X and Y
• Miller’s Jackknife test of equal dispersion between X and Y
• Lepage rank test of equal location and dispersion between X and Y
• Kolmogorov-Smirnov test for general differences between X and Y
Summarise and explain the results of the five tests.
(b) If evident of differences between X and Y is found in part (a), determine whether a
linear transform can remove such differences; in other words, aX b+ and Y no longer
have the differences between X and Y for some constants a and b .
If such a and b are identified, repeat part (a) on aX b+ and Y to verify the results and
specify the differences between X and Y .
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Question 5 [8 marks]
The following data in a one-way layout were recorded from an experiment:
Treatment
1 2 3 4 5 6
40 60 78 60 58 23
23 54 85 82 35 65
58 51 45 54 48 28
(a) Test 0 1 6:H τ τ= = against the umbrella alternatives 1 1 2 3 4 5 6:H τ τ τ τ τ τ≤ ≤ ≥ ≥ ≥ with
at least one strict inequality for a known peak 3p = . Indicate the strength of evidence
to support 1H by the exact p-value of the test using R.
(b) If the peak of the umbrella alternatives is unknown, calculate the Mack-Wolfe test
statistic *pˆA for unknown peak and find an approximate p-value of the test using R.
Compare the results of parts (a) and (b).