Copyright c 2012 iowa state university statistics 511

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Unformatted text preview: s 511 10 / 31 Note that w= = = = = Ay A (Xβ + ) A Xβ + A 0+A A Thus, d w = A ∼ N (A 0, A ΣA) = N (0, A ΣA), and the distribution of w depends on γ but not β . Copyright c 2012 (Iowa State University) Statistics 511 11 / 31 The log likelihood function in this case is 1 n−r 1 log(2π ). (γ |w) = − log |A ΣA|− w (A ΣA)−1 w− 2 2 2 ˆ An MLE for γ , say γ , can be found in the general case using numerical methods to obtain the REML estimate of γ . Copyright c 2012 (Iowa State University) Statistics 511 12 / 31 In 6ll, we take the time to prove that every set of n − r linearly independent error contrasts yields the same REML estimator of γ . As an example, consider the special case where i.i.d. y1 , . . . , yn ∼ N (µ, σ 2 ). Then X = 1, β = µ, and Σ = σ 2 I. Copyright c 2012 (Iowa State University) Statistics 511 13 / 31 It follows that a1 a2 an−1 = (1, −1, 0, 0, . . . , 0) = (0, 1, −1, 0, . . . , 0) . . . = (0, 0, . . . , 0, 1, −1) and Copyright c 2012 (Iowa State University) Statistics 511 14 / 31 b1 b2 bn−1 = (1, 0, 0, . . . , 0, −1) = (0, 1, 0, . . . , 0, −1) . . . = (0, 0, . . . , 0, 1, −1) are each a set of n − r = n − 1 linear independent vectors that can be used to form error contrasts. Copyright c 2012 (Iowa State University) Statistics 511 15 / 31 Either a1 y y1 − y2 a2 y y2 − y3 w= . = . . . . . an−1 y yn−1 − yn b1 y y1 − yn b2 y y2 − yn or v = . = . . . . . bn−1 y yn−1 − yn could be used to obtain the same REML estimator of σ 2 . Copyright c 201...
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