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Inference for the mean vector

Inference for the mean vector. Univariate Inference. Let x 1 , x 2 , … , x n denote a sample of n from the normal distribution with mean m and variance s 2 . Suppose we want to test H 0 : m = m 0 vs H A : m ≠ m 0 The appropriate test is the t test:. The test statistic:.

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Inference for the mean vector

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  1. Inference for the mean vector

  2. Univariate Inference Let x1, x2, … , xn denote a sample of n from the normal distribution with mean mand variance s2. Suppose we want to test H0: m= m0 vs HA: m≠ m0 The appropriate test is the t test: The test statistic: Reject H0 if |t| > ta/2

  3. The multivariate Test Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Suppose we want to test

  4. Roy’s Union- Intersection Principle This is a general procedure for developing a multivariate test from the corresponding univariate test. • Convert the multivariate problem to a univariate problem by considering an arbitrary linear combination of the observation vector.

  5. Perform the test for the arbitrary linear combination of the observation vector. • Repeat this for all possible choices of • Reject the multivariate hypothesis if H0 is rejected for any one of the choices for • Accept the multivariate hypothesis if H0 is accepted for all of the choices for • Set the type I error rate for the individual tests so that the type I error rate for the multivariate test is a.

  6. Application of Roy’s principle to the following situation Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Suppose we want to test Then u1, …. un is a sample of n from the normal distribution with mean and variance .

  7. to test we would use the test statistic:

  8. and

  9. Thus We will reject if

  10. Using Roy’s Union- Intersection principle: We will reject We accept

  11. i.e. We reject We accept

  12. Consider the problem of finding: where

  13. thus

  14. Thus Roy’s Union- Intersection principle states: We reject We accept is called Hotelling’s T2 statistic

  15. Choosing the critical value for Hotelling’s T2 statistic We reject , we need to find the sampling distribution of T2 when H0 is true. It turns out that if H0 is true than has an F distribution with n1= p and n2= n - p

  16. ThusHotelling’s T2 test We reject or if

  17. Another derivation of Hotelling’s T2 statistic Another method of developing statistical tests is the Likelihood ratio method. Suppose that the data vector, , has joint density Suppose that the parameter vector, , belongs to the set W. Let wdenote a subset of W. Finally we want to test

  18. The Likelihood ratio test rejects H0 if

  19. The situation Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Suppose we want to test

  20. The Likelihood function is: and the Log-likelihood function is:

  21. the Maximum Likelihood estimators of are and

  22. the Maximum Likelihood estimators of when H 0 is true are: and

  23. The Likelihood function is: now

  24. Thus similarly

  25. and

  26. Note: Let

  27. and Now and

  28. Also

  29. Thus

  30. Thus using

  31. Then Thus to reject H0 if l< la This is the same as Hotelling’s T2 test if

  32. Example For n = 10 students we measure scores on • Math proficiency test (x1), • Science proficiency test (x2), • English proficiency test (x3) and • French proficiency test (x4) The average score for each of the tests in previous years was 60. Has this changed?

  33. The data

  34. Summary Statistics

  35. Simultaneous Inference for means Recall (Using Roy’s Union Intersection Principle)

  36. Now

  37. Thus and the set of intervals Form a set of (1 – a)100 % simultaneous confidence intervals for

  38. Recall Thus the set of (1 – a)100 % simultaneous confidence intervals for

  39. The two sample problem

  40. Univariate Inference Let x1, x2, … , xn denote a sample of n from the normal distribution with mean mx and variance s2. Let y1, y2, … , ym denote a sample of n from the normal distribution with mean my and variance s2. Suppose we want to test H0: mx= myvs HA: mx≠ my

  41. The appropriate test is the t test: The test statistic: Reject H0 if |t| > ta/2 d.f. = n + m -2

  42. The multivariate Test Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Let denote a sample of m from the p-variate normal distribution with mean vector and covariance matrix S. Suppose we want to test

  43. Hotelling’s T2 statisticfor the two sample problem if H0 is true than has an F distribution with n1= p and n2= n +m – p - 1

  44. ThusHotelling’s T2 test We reject

  45. Simultaneous inference for the two-sample problem • Hotelling’s T2 statistic can be shown to have been derived by Roy’s Union-Intersection principle

  46. Thus

  47. Thus

  48. Thus Hence

  49. Thus form 1 – a simultaneous confidence intervals for

  50. Hotelling’s T2 test A graphical explanation

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