ACST8040 Quantitative Research Methods Assignment 2    Question 1 [20 marks] An insurance company has installed a new measure to reduce the cost on a line of business

ACST8040 Quantitative Research Methods

Assignment 2   

Question 1 [20 marks]

An insurance company has installed a new measure to reduce the cost on a line of business. The costs (in $m) of such a business line in 10 branches of the company before and after the new measure is taken over a certain period of time are recorded in the table below: 

Let X and Y represent the costs before and after the new measure takes effect, respectively, and ? the median of the cost reduction.  

Based on the data provided in the above table, perform the following analyses:

1. Calculate the value of the Wilcoxon signed rank test statistic T^? based on the difference

Y?X between the costs before and after the new measure takes effect.           [4]

2. Determine if there is sufficient evidence at the 5% level that the new measure is effective to reduce the cost by the Wilcoxon signed rank test using the exact p-value conditional on any ties by enumeration.           [7]
3. Repeat the question in part (b) by the normal approximation.  [3]
4. Estimate the median ? and obtain its exact confidence interval with a target at least 95% confidence level based on the Wilcoxon signed ranks. [6]

Question 2 [15 marks]

Prove the following results on the Wilcoxon signed-rank test statistic T^? for one-sample location and the Ansari-Bradley rank test statistic C for two-sample dispersion:

1. Let X₁ and X₂ be two independent continuous random variables. If X₁ ~?X₂ with a common median 0, then T^??S?I_{X1?_0}?2I_{X2?_0},  where “~” represents “having the same distribution”. You may use the following equation with density f₁(x) of X₁: 

?

Pr(0? X₁??X₂)^?? Pr(?X₂ ? x) f₁ (x)dx   [8]

0

2. If the total size N of two samples combined is even and assume no ties, then C has a symmetric distribution about E₀[C] under H₀ :?² ?1.  [7]

Hint: For each (r₁,?,r_n) drawn from {a₁,?,a_N}??1,2,?,N 2?1,N 2,N 2?1,?,2,1? as Y-scores, consider (r?1,?,r?n) with r?i ?N 2?1?r_i , i ?1,?,n. 

Question 3 [25 marks]

The following X and Y represent two independent samples on the profits (in $m) of two banks in different areas during a financial year:

X?(35,57,39,30,52,42,38,49,24,36,32,44) 

Y?(47,40,61,80,28,89,54,74,45,50,21)

Carry out the following statistical analyses based on the samples X and Y :

1. Under the location-shift model, use the Wilcoxon rank sum test to determine whether there is sufficient of evidence for sample Y to have a greater location parameter than sample X by the normal approximation at the 5% level of significance. [4]
2. Under the location-scale parameter model, test the null hypothesis H₀:Var(X)?Var(Y) against the alternative H₁:Var(X)?Var(Y) by the Ansari-Bradley rank test with the normal approximation at the 10% level of significance.  [4]
3. Calculate the values of A₁,A₂,?,A_m and B₁,B₂,?,B_n for the Miller’s Jackknife test. Then find the approximate p-value for Var(X)?Var(Y) by the Miller’s Jackknife test and draw a conclusion.        [5]
4. Let X^*?3X?(3X₁,?,3X_m) and Y^*?Y?66?(Y₁?66,?,Y_n?66). 

Find the empirical distribution functions F₁₂^* (t) of X^* and G₁₁^* (t) of Y^* at the ordered values Z₍₁₎ ???Z₍₂₃₎ of ?X^*,Y^*?. Then calculate the value of the test statistic J of the two-sample Kolmogorov-Smirnov test for general differences between X^* and Y^*.

Verify the value of J and find the p-value of the test by R.     [5]

5. Based on the results in parts (a) – (d), answer the following questions with reasoning:
   1. What conclusions can be drawn from part (d)?
   2. What relations in the locations and dispersions between X and Y can be inferred from the analyses?
   3. Is the location-shift model appropriate in part (a)?
   4. Is the location-scale parameter model justified in parts (b) and (c)?
   5. Are the results in parts (a) – (c) justified?
   6. What observations can be made from the analyses for comparing the profitability and profit stability of the two banks?       [7] Question 4 [20 marks]

A set of data {X_ij} in a one-way layout with 5 treatments are listed below:

Treatment

Denote the effects of treatments 1,?,5 by ^?₁^,?,^?₅ respectively.

1. Test the null hypothesis ^H₀ :?₁????₅ against general alternatives at the 5% level of significance by the Kruskal-Wallis test with the approximate rejection rule. [5] 
2. Test H₀ :?₁????₅ against ordered alternatives H₁:?₁????₅ with at least one strict inequality at the 1% level of significance by the Jonckheere-Terpstra test with the normal approximation.  [6]
3. Determine whether ?_u ??₁ or ?_u ??₁ for u? 2,3,4,5 by the Nemenyi-Damico-Wolfe one-sided treatments-versus-control multiple comparison procedure at ??10%. You can use R to find the critical point for making the decisions. [4] 
4. Comment on the differences in parts (a) – (c) regarding the decisions on treatment effects

?₁,?,?₅ . Are they consistent or contradictive? Explain why or why not.       [5]

Question 5 [20 marks]

A sample of data in a one-way layout with four treatments are presented below:

Let ?₁,?,?₄ denote the effects of the 4 treatments. 

1. Test H₀ :?₁????₄ against the alternative H₁:?₁??₂ ??₃??₄ with at least one strict inequality by the Mack-Wolfe test with known peak p? 2 at the 5% level using the large-sample normal approximation.          [7]
2. If the peak p is unknown, find an estimate pˆ of p and calculate the Mack-Wolfe test statistic A^*_pˆ with unknown peak.     [5]
3. Test ^H₀ :?₁????₄ against umbrella alternatives with unknown peak p at the 5% level of significance using R. [4]
4. If the estimate pˆ of p in part (b) were assumed known, what would be the result of the Mack-Wolfe test with known peak pˆ ? If the test result is different from that of part (c), explain the main reason for the difference.  [4]