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Unformatted text preview: PSTAT 120C: Solutions to Assignment # 8 June 7, 2006 1. (a) In the case where all the r ij = r , y i = 1 br b X j =1 r X m =1 y ijm and y = 1 bkr b X j =1 k X i =1 r X m =1 y ijm The expected value of each observation is E ( y ijk ) = + i + j , and so the expected value of the overall mean is E ( y ) = 1 bkr b X j =1 k X i =1 R X m =1 ( + i + j ) = 1 bkr b X j =1 k X i =1 r ( + i + j ) = 1 bk b X j =1 k X i =1 ( + i + j ) = + 1 k k X i =1 i + 1 b b X j =1 j = because, by definition k i =1 i = b j =1 j = 0. Similarly, E ( y i ) = 1 br b X j =1 r X m =1 ( + i + j ) = 1 b b X j =1 ( + i + j ) = + i + 1 b b X j =1 j = + i . Therefore, E ( i ) = + i = i , and this estimator is unbiased. 1 (b) When the r ij are different, we can write the estimator as i = 1 b j =1 r ij b X j =1 r i,j X m =1 y ijk 1 k i =1 b j =1 r ij k X i =1 b X...
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 Spring '07
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