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Inferences about a Mean Vector

This article explores the concepts of making inferences about mean vectors in multivariate statistics. From understanding student's t-distribution and F-distribution to testing hypotheses and constructing confidence intervals, the text delves into evaluating T2 statistics and likelihood ratio tests. It covers topics such as confidence regions, ellipsoids, and simultaneous confidence statements. Examples and methods like Bonferroni inequality and multiple comparisons are detailed to enhance comprehension. The goal is to provide a comprehensive guide on reaching valid conclusions about population means based on sample data.

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Inferences about a Mean Vector

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

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

  3. Plausibility of m0 as a Value for a Normal Population Mean

  4. Student’s t-distribution

  5. Student’s t-distribution

  6. Test of Hypothesis

  7. Confidence Interval

  8. Plausibility of m0 as a Multivariate Normal Population Mean

  9. T2 as an F-Distribution

  10. F-Distribution

  11. F-Distribution

  12. Nature of T2-Distribution

  13. Test of Hypothesis

  14. Example 5.1 Evaluating T2

  15. Example 5.2 Testing aMean Vector

  16. Invariance of T2-Statistic

  17. T2-Statistic from Likelihood Ratio Test

  18. T2-Statistic from Likelihood Ratio Test

  19. Result 4.10

  20. Likelihood Ratio Test

  21. Result 5.1

  22. Proof of Result 5.1

  23. Proof of Result 5.1

  24. Computing T2 from Determinants

  25. General Likelihood Ratio Method

  26. Result 5.2

  27. 100(1-a)% Confidence Region

  28. 100(1-a)% Confidence Region

  29. Axes of the Confidence Ellipsoid

  30. Example 5.3 :Microwave Oven Radiation

  31. Example 5.3 :95% Confidence Region

  32. Example 5.3 : 95% Confidence Ellipse for m

  33. Example 5.3 : 95% Confidence Ellipse for m

  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. Concept of Simultaneously Confidence Statements

  36. Confidence Interval of Linear Combination of Variables

  37. Maximum t2 Value for All a

  38. Maximization Lemma

  39. Result 5.3: T2 Interval

  40. Comparison of t- and T2-Intervals

  41. Simultaneous T2-Intervals

  42. Example 5.4: Shadows of the Confidence Ellipsoid

  43. Example 5.5

  44. Example 5.5

  45. Example 5.5: Confidence Ellipses for Pairs of Means

  46. One-at-a-Time Intervals

  47. Bonferroni Inequality

  48. Bonferroni Method of Multiple Comparisons

  49. Example 5.6

  50. Example 5.6

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