 # Inference for the mean vector Univariate Inference Let

• Slides: 63 Inference for the mean vector Univariate Inference Let x 1, x 2, … , xn 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 HA : m ≠ m 0 The appropriate test is the t test: The test statistic: Reject H 0 if |t| > ta/2 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 Roy’s Union- Intersection Principle This is a general procedure for developing a multivariate test from the corresponding univariate test. 1. Convert the multivariate problem to a univariate problem by considering an arbitrary linear combination of the observation vector. 2. 3. 4. 5. 6. Perform the test for the arbitrary linear combination of the observation vector. Repeat this for all possible choices of Reject the multivariate hypothesis if H 0 is rejected for any one of the choices for Accept the multivariate hypothesis if H 0 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. 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 u 1, …. un is a sample of n from the normal distribution with mean and variance. to test we would use the test statistic: and Thus We will reject if Using Roy’s Union- Intersection principle: We will reject We accept i. e. We reject We accept Consider the problem of finding: where thus Thus Roy’s Union- Intersection principle states: We reject We accept is called Hotelling’s T 2 statistic Choosing the critical value for Hotelling’s T 2 statistic We reject , we need to find the sampling distribution of T 2 when H 0 is true. It turns out that if H 0 is true than has an F distribution with n 1 = p and n 2 = n - p Thus Hotelling’s T 2 test We reject or if Another derivation of Hotelling’s T 2 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 w denote a subset of W. Finally we want to test The Likelihood ratio test rejects H 0 if 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 The Likelihood function is: and the Log-likelihood function is: the Maximum Likelihood estimators of are and the Maximum Likelihood estimators of when H 0 is true are: and The Likelihood function is: now Thus similarly and Note: Let and Now and Also Thus Thus using Then Thus to reject H 0 if l < la This is the same as Hotelling’s T 2 test if Example For n = 10 students we measure scores on – Math proficiency test (x 1), – Science proficiency test (x 2), – English proficiency test (x 3) and – French proficiency test (x 4) The average score for each of the tests in previous years was 60. Has this changed? The data Summary Statistics Simultaneous Inference for means Recall (Using Roy’s Union Intersection Principle) Now Thus and the set of intervals Form a set of (1 – a)100 % simultaneous confidence intervals for Recall Thus the set of (1 – a)100 % simultaneous confidence intervals for The two sample problem Univariate Inference Let x 1, x 2, … , xn denote a sample of n from the normal distribution with mean mx and variance s 2. Let y 1, y 2, … , ym denote a sample of n from the normal distribution with mean my and variance s 2. Suppose we want to test H 0: mx = my vs HA : mx ≠ my The appropriate test is the t test: The test statistic: Reject H 0 if |t| > ta/2 d. f. = n + m -2 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 Hotelling’s T 2 statistic for the two sample problem if H 0 is true than has an F distribution with n 1 = p and n 2 = n +m – p - 1 Thus Hotelling’s T 2 test We reject Simultaneous inference for the two-sample problem • Hotelling’s T 2 statistic can be shown to have been derived by Roy’s Union-Intersection principle Thus Thus Thus Hence Thus form 1 – a simultaneous confidence intervals for Hotelling’s T 2 test A graphical explanation Hotelling’s T 2 statistic for the two sample problem is the test statistic for testing: Hotelling’s T 2 test X 2 Popn A Popn B X 1 X 2 Univariate test for X 1 Popn A Popn B X 1 X 2 Univariate test for X 2 Popn A Popn B X 1 X 2 Univariate test for a 1 X 1 + a 2 X 2 Popn A Popn B X 1 Mahalanobis distance A graphical explanation Euclidean distance Mahalanobis distance: S, a covariance matrix Hotelling’s T 2 statistic for the two sample problem Case I X 2 Popn A Popn B X 1 Case II X 2 Popn A Popn B X 1 In Case I the Mahalanobis distance between the mean vectors is larger than in Case II, even though the Euclidean distance is smaller. In Case I there is more separation between the two bivariate normal distributions