Contents
Abstract 2
1. Introduction 2
2. Inference under Posterior Predictive Sampling 4
2.1. Single Imputation Case 4
2.2. Multiple Imputation Case 5
3. Further Inference under Multiple Imputation 5
3.1. Confidence Intervals Based on Graybill-Deal estimator 5
3.2. Confidence Intervals Based on the Overall Mean 9
3.3. Reiter's Approximation 10
4. Exact Tests under Multiple Imputation 11
4.1. Exact Tests Based on the GD Estimator & Overall Mean 11
4.2. Exact Tests Based on P-Values 12
5. Local Powers of the P-Value Based Tests 13
5.1. Local Power of Tippett's Test (LP1) 16
5.2. Local Power of Inverse Normal Test (LP2) 16
5.3. Local Power of Fisher's Inverse Chi-square Test (LP3) 17
6. Simulation Studies 17
6.1. Comparison of Confidence Intervals 17
6.2. Comparison of the Exact Tests 19
7. Privacy Protection 21
7.1. Single Imputation Case 21
7.2. Multiple Imputation Case 22
8. A Real Life Data 24
9. Concluding Remarks 25
Availability of Data and Materials 26
Acknowledgements 26
Declarations 26
Annexure 26
References 28
Table 1. TABLE SHOWING THE EXPECTED LENGTHS OF DIFFERENT CI's UNDER DIFFERENT CHOICES OF m and n (α = 4, γ = 0.05) 17
Table 2. TABLE SHOWING THE TRUE COVERAGE PROBABILITIES (TCP) OF DIFFERENT CI's WITH THEIR AVERAGE LENGTHS (AL) UNDER DIFFERENT CHOICES... 18
Table 3. TABLE SHOWING THE FALSE COVERAGE PROBABILITIES FOR DIFFERENT CI's UNDER DIFFERENT CHOICES OF /μ/ (m = 5, α = 4, γ = 0.05) 18
Table 4. TABLE SHOWING THE FALSE COVERAGE PROBABILITIES FOR DIFFERENT CI's UNDER DIFFERENT CHOICES OF /μ/ (m = 10, α = 4, γ = 0.05) 19
Table 5. TABLE SHOWING THE CUTOFF POINTS FOR P-VALUE BASED TESTS UNDER DIFFERENT CHOICES OF m and n (α = 4, γ = 0.05) 20
Table 6. TABLE SHOWING THE LOCAL POWERS FOR DIFFERENT EXACT TESTS UNDER DIFFERENT CHOICES OF m, n and δ (α = 4, γ = 0.05) 20
Table 7. TABLE SHOWING THE PRIVACY MEASURES FOR DIFFERENT CHOICES OF m & n (ϵ = 0.001 and α = 4) UNDER SINGLE IMPUTATION CASE 23
Table 8. TABLE SHOWING THE PRIVACY MEASURES FOR DIFFERENT CHOICES OF m & n (ϵ = 0.001 and α = 4) UNDER MULTIPLE IMPUTATION CASE 23
Table 9. TABLE SHOWING THE SUMMARY STATISTICS FOR EACH IMPUTATION UNDER PPS (m = 5 and α = 4) 24
Table 10. TABLE SHOWING THE UPPER & LOWER BOUNDS OF THE SEVEN CI's (α = 4, 1 - γ = 0.95) 25
Figure 1. Figure showing the Local Powers of Exact Tests under m = 10 and n = 25 21