1. 1 Inferences about a Mean Vector Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia

2. 2 Inference Reaching valid conclusions concerning a population on the basis of information from a sample

3. 3 Plausibility of  0 as a Value for a Normal Population Mean

4. 4 Student’s t-distribution

5. 5

6. 6 Test of Hypothesis

7. 7 Confidence Interval

8. 8 Plausibility of  0 as a Multivariate Normal Population Mean

9. 9 T 2 as an F-Distribution

10. 10 F-Distribution

11. 11 F-Distribution

12. 12 Nature of T 2 -Distribution

13. 13 Test of Hypothesis

14. 14 Example 5.1 Evaluating T 2

15. 15 Example 5.2 Testing a Mean Vector

16. 16 Invariance of T 2 -Statistic

17. 17 T 2 -Statistic from Likelihood Ratio Test

18. 18 T 2 -Statistic from Likelihood Ratio Test

19. 19 Result 4.10

20. 20 Likelihood Ratio Test

21. 21 Result 5.1

22. 22 Proof of Result 5.1

23. 23 Proof of Result 5.1

24. 24 Computing T 2 from Determinants

25. 25 General Likelihood Ratio Method

26. 26 Result 5.2

27. 27 100(1-  )% Confidence Region

28. 28 100(1-  )% Confidence Region

29. 29 Axes of the Confidence Ellipsoid

30. 30 Example 5.3 : Microwave Oven Radiation

31. 31 Example 5.3 : 95% Confidence Region

32. 32 Example 5.3 : 95% Confidence Ellipse for 

33. 33 Example 5.3 : 95% Confidence Ellipse for 

34. 34 Simultaneous Confidence Statements Sometimes we need confidence statements about the individual component means All if the separate confidence statements should hold simultaneously with a specified high probability

35. 35 Concept of Simultaneously Confidence Statements

36. 36 Confidence Interval of Linear Combination of Variables

37. 37 Maximum t 2 Value for All a

38. 38 Maximization Lemma

39. 39 Result 5.3: T 2 Interval

40. 40 Comparison of t - and T 2 -Intervals

41. 41 Simultaneous T 2 -Intervals

42. 42 Example 5.4: Shadows of the Confidence Ellipsoid

43. 43 Example 5.5

44. 44 Example 5.5

45. 45 Example 5.5: Confidence Ellipses for Pairs of Means

46. 46 One-at-a-Time Intervals

47. 47 Bonferroni Inequality

48. 48 Bonferroni Method of Multiple Comparisons

49. 49 Example 5.6

50. 50 Example 5.6

51. 51 (Length of Bonferroni Interval )/ (Length of T 2 -Interval)

52. 52 Limit Distribution of Statistical Distance

53. 53 Result 5.4

54. 54 Result 5.5

55. 55 Result 5.5

56. 56 Example 5.7: Musical Aptitude Profile for 96 Finish Students

57. 57 Example 5.7: Simultaneous 90% Confidence Limits

58. 58 One-at-a-Time and Bonferroni Confidence Intervals

59. 59 Large-Sample 95% Intervals for Example 5.7

60. 60 95% Intervals for Example 5.7

61. 61 Control Chart Represents collected data to evaluate the capabilities and stability of the process Identify occurrences of special causes of variation that come from outside of the usual process

62. 62 Example 5.8: Overtime Hours for a Police Department

63. 63 Example 5.8 Univariate Control Chart

64. 64 Monitoring a Sample for Stability

65. 65 Example 5.9: 99% Ellipse Format Chart

66. 66 Example 5.9: -Chart for X 2

67. 67 Example 5.10: T 2 Chart for X 1 and X 2

68. 68 Example 5.11: Robotic Welders

69. 69 Example 5.11: T 2 Chart

70. 70 Example 5.11: 99% Quality Control Ellipse for ln(Gas flow) and voltage

71. 71 Example 5.11: -Chart for ln(Gas flow)

72. 72 Control Regions for Future Individual Observations Set for future observations from collected data when process is stable Forecast or prediction region –in which a future observation is expected to lie

73. 73 Result 5.6

74. 74 Proof of Result 5.6

75. 75 Result 4.8

76. 76 Example 5.12 Control Ellipse

77. 77 T 2 -Chart for Future Observations

78. 78 Control Chart Based on Subsample Means

79. 79 Control Chart Based on Subsample Means

80. 80 Control Regions for Future Subsample Observations

81. 81 Control Chart Based on Subsample Means

82. 82 EM Algorithm Prediction step –Given some estimate of the unknown parameters, predict the contribution of the missing observations to the sufficient statistics Estimation step –Use the predicted statistics to compute a revised estimate of the parameters Cycle from one step to the other

83. 83 Complete-Data Sufficient Statistics

84. 84 Prediction Step for Multivariate Normal Distribution

85. 85 Result 4.6

86. 86 Estimation Step for Multivariate Normal Distribution

87. 87 Example 5.13: EM Algorithm

88. 88 Example 5.13: Prediction Step

89. 89 Example 5.13: Prediction Step

90. 90 Example 5.13: Prediction Step

91. 91 Example 5.13: Estimation Step

92. 92 Time Dependence in Observations

93. 93 Coverage Probability of the 95% Confidence Ellipsoid